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Weak Elliptic Maximum Principles and Holder Regularity

1 · Prerequisites

2 · Summary

This page develops the weak maximum principle for coercive divergence-form equations with bounded measurable coefficients, the De Giorgi level-set iteration that proves it, and the De Giorgi--Nash--Moser interior regularity theory built on the same Caccioppoli and level-set machinery. The principal coefficients are real, measurable, bounded and uniformly elliptic; the De Giorgi and Harnack items additionally assume symmetry with $θ I \leq A \leq M_{a}^{2} I$ , while the maximum principle uses the coefficientwise bounds of its operator definition; the principal operator is $L_{0} u = - D_{i} (a^{i j} D_{j} u)$ with its sesquilinear form $a_{0}$ , and lower-order terms $b, c$ appear only in the maximum-principle items under explicit sign hypotheses. The homogeneous weak maximum principle assumes $c \geq 0$ a.e. together with the weak sign condition $\int (c ζ + b^{i} D_{i} ζ) \geq 0$ for nonnegative test functions $ζ$ , the weak form of $c - div b \geq 0$ ; the forcing case assumes $b \equiv 0$ , $c \geq 0$ and an $L^{q}$ source with $q > n / 2$ , whose maximum bound adds $C ∥ f^{+} ∥_{L^{q}}$ to the positive boundary supremum; the local forcing estimates explicitly carry the scale factor $R^{2 - n / q}$ . The truncated positive part is an admissible test, the truncated Caccioppoli inequality and the Sobolev level-set iteration are recorded for arbitrary levels, and the local boundedness, oscillation reduction, Holder regularity, Moser iteration, weak Harnack and Harnack theorems follow with constants depending only on $n, q, θ, M_{a}$ and each estimate's stated exponent and radius ratios. The zero-set propagation and the strong maximum principle are recorded for $n \geq 3$ on connected open sets and close the page, and the superscript conventions of the trace and of the boundary supremum are those of the weak-subsolution definition.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Weak subsolutions and supersolutions of a divergence-form equation

Definition

Assume Countable Choice and the Axiom of Choice for the Sobolev, trace and embedding interfaces below. Let $n \geq 1$ , let $Ω \subset R^{n}$ be open, and let $L$ and its sesquilinear form $a$ be as in Uniformly elliptic divergence-form operators and their sesquilinear forms with ellipticity constant $θ$ and coefficient bounds $M_{a}, M_{b}, M_{c}$ . For the order comparison below, take real coefficients and a real-valued source $f \in L_{l o c}^{1} (Ω)$ (Locally integrable functions as regular distributions, The space $L^{p} (μ)$ as the quotient by null functions).

A real class $u \in H^{1} (Ω; R)$ (The notation $H^{k}$ and the reserved zero-boundary symbol) is a local weak subsolution of $L u = f$ on $Ω$ if $a (u, φ) \leq \int_{Ω} f φ d x for every nonnegative φ \in C_{c}^{\infty} (Ω; R),$ and a local weak supersolution if the reverse inequality holds; it is a local weak solution if equality holds for every real $φ \in C_{c}^{\infty} (Ω)$ . These tests make every pairing finite for $f \in L_{l o c}^{1}$ .

Global Sobolev-test version. If the source defines a continuous functional $F \in H^{- 1} (Ω) : = (H_{0}^{1} (Ω))^{*}$ (The negative Sobolev space $H^{- 1} (Ω)$ ), then a real $u \in H^{1} (Ω)$ is a global weak subsolution if $a (u, v) \leq F (v) for every nonnegative v \in H_{0}^{1} (Ω),$ with the reverse inequality defining a global weak supersolution and equality defining a global weak solution. The local and global formulations agree when both apply, by continuity of the form and $F$ and the following positive-cone density argument. Given $0 \leq v \in H_{0}^{1}$ , choose real $z_{j} \in C_{c}^{\infty} (Ω)$ converging to $v$ in $H^{1}$ and a subsequence converging a.e.; such a subsequence follows by choosing $∥ z_{j} - v ∥_{2}^{2} \leq 2^{- 3 j}$ and applying Chebyshev and countable subadditivity to ${∣ z_{j} - v ∣ > 2^{- j}}$ . The positive-part chain rule gives $D z_{j}^{+} - D v = 1_{{z_{j} > 0}} (D z_{j} - D v) + (1_{{z_{j} > 0}} - 1_{{v > 0}}) D v$ . The second term tends to zero in $L^{2}$ by dominated convergence, since its indicator converges where $v > 0$ and $D v = 0$ a.e. where $v = 0$ ; the first term and the function difference converge in $L^{2}$ . Thus $z_{j}^{+} \to v$ in $H^{1}$ . Each $z_{j}^{+}$ has compact support, so zero extension followed by nonnegative unit-mass mollification with sufficiently small radius gives a nonnegative $C_{c}^{\infty} (Ω)$ approximant within $1 / j$ in $H^{1}$ (Positive, negative, and truncated Sobolev functions, Compactly supported Sobolev functions extend by zero in every integer order, Local smooth approximation in integer-order Sobolev spaces, Dominated convergence, Chebyshev-Markov inequality for the integral). In particular, $f \in L^{2} (Ω)$ suffices by Cauchy--Schwarz and $H_{0}^{1} ↪ L^{2}$ ; on a bounded $C^{1}$ domain, $f \in L^{q} (Ω)$ with $q > n / 2$ for $n \geq 3$ (or $q > 1$ for $n = 2$ ) suffices because $q^{'}$ lies in the Sobolev range $H_{0}^{1} ↪ L^{q^{'}}$ (The Sobolev inequality for zero-boundary Sobolev closures on open sets, The critical Sobolev embedding into every finite $L^{q}$ , Holder's inequality for integrals, including the endpoint cases). If $f \in L_{l o c}^{2} (Ω)$ , the local inequality also extends to nonnegative $H_{0}^{1} (U)$ tests on any bounded open $U ⋐ Ω$ , since $f ∣_{U} \in L^{2} (U)$ .

For complex-valued coefficients or classes, only the weak-solution identity with a specified continuous complex source functional is used; no subsolution or supersolution order is defined by comparing complex numbers. In the real setting, a weak solution is both a subsolution and a supersolution exactly when the same source functional is used in both inequalities.

Signed essential extrema. For a real measurable class on a positive-measure set $E$ , write ${ess sup}_{E} u : = \inf {t \in R : u \leq t a.e. on E}$ in the extended reals, and ${ess inf}_{E} u : = - {ess sup}_{E} (- u)$ . A finite essential supremum $s$ is itself an a.e. upper bound: take the union of the null exceptional sets for the bounds $s + 1 / j$ . These signed extrema differ from the essential supremum of $∣ u ∣$ used to define the $L^{\infty}$ norm. For a continuous representative on an open set, its pointwise and essential extrema agree, since a strict violation of an essential bound would hold on a nonempty open set of positive measure.

Weak boundary order (for $n \geq 2$ ). Let in addition $n \geq 2$ and let $Ω$ be a bounded $C^{1}$ domain (Bounded C^k domains and boundary charts) with trace operator $T$ (The $L^{p}$ trace operator on a bounded $C^{1}$ domain). For real $u, v \in H^{1} (Ω)$ and $k \in R$ one writes $u \leq k$ on $\partial Ω$ if $(u - k)^{+} \in H_{0}^{1} (Ω)$ , and $u \leq v$ on $\partial Ω$ if $(u - v)^{+} \in H_{0}^{1} (Ω)$ ; by The kernel of the trace is the closure of the test functions these are respectively the statements $T u \leq k$ and $T u \leq T v$ a.e. on $\partial Ω$ , as justified by A function whose trace is at most a level has positive part in the zero-boundary space ↗, and it is independent of the chosen representatives. The boundary supremum is $\sup_{\partial Ω} u : = \inf {k \in R : u \leq k on \partial Ω} \in R \cup {+ \infty},$ with $\inf \emptyset : = + \infty$ ; the set is nonempty as soon as $u$ is essentially bounded above.

Conventions

Sign convention. In the real order theory, the subsolution inequality is $a (u, v) \leq \int_{Ω} f v$ against nonnegative tests. For the operator $L = - D_{i} (a^{i j} D_{j}) + b^{i} D_{i} + c$ , the favourable pointwise sign in the maximum principle is $c \geq 0$ ; negating a supersolution preserves the same coefficients, so the same sign is favourable for the corresponding minimum estimate.
Real order versus complex identities. The maximum-principle, De Giorgi and Harnack results use real-valued $u$ , real coefficients and real sources, so their inequalities compare real numbers. Complex local weak solutions use the compactly supported identity of Local weak solutions of a divergence-form operator; a complex global weak identity uses a specified continuous functional on $H_{0}^{1}$ . No order notion is assigned to a complex-valued form.
No boundary condition is imposed by the subsolution or supersolution notion itself, and the boundary order is only introduced on a bounded $C^{1}$ domain, through the trace; it is never read off pointwise boundary values of a class.

Sources

Simon, Lectures on Partial Differential Equations, Lecture 13, printed pp. 147-158: the real weak form against nonnegative $φ \in C_{c}^{\infty}$ , the conventions (i)-(iv) for $u \leq 0$ on $\partial Ω$ and $\sup_{\partial Ω} u = \inf {k : u \leq k on \partial Ω}$ , and the weak maximum principle Theorem 4. Simon works with real-valued data; the present definition records the local real order convention and the separate $H^{- 1}$ global extension.
Teschl, PDE: From Classical to Modern, Chapter 10 Section 1: Theorem 10.1, Lemma 10.2 and the same boundary convention $(v - u)^{+} \in H_{0}^{1} (U)$ for $v \leq u$ on $\partial U$ .
Schikorra, Partial Differential Equations, Chapter 2 Sections II.1-II.2: Definitions II.1.1 and II.1.3, the sign convention for the zeroth-order term, and Theorem II.2.1.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Positive-part truncation calculus and admissible cut-off weak tests

Statement

Assume Countable Choice together with the Axiom of Choice, inherited through the published ACL characterisation and the chain-rule interfaces cited below. Let $n \geq 1$ , let $Ω \subseteq R^{n}$ be open, let $u \in H^{1} (Ω; R)$ and $k \in R$ , and put $u_{k} : = (u - k)^{+}$ and $u_{k}^{-} : = (u - k)^{-}$ on measurable representatives. Then $u_{k}, u_{k}^{-} \in H_{l o c}^{1} (Ω; R)$ with $D u_{k} = 1_{{u > k}} D u, D u_{k}^{-} = - 1_{{u < k}} D u a.e. on Ω,$ and $D u_{k} = 0$ a.e. on ${u \leq k}$ . If $∣ Ω ∣ < \infty$ , then $u_{k}, u_{k}^{-} \in H^{1} (Ω)$ ; more generally, either truncation belongs to $H^{1} (Ω)$ whenever that truncation is in $L^{2} (Ω)$ . In particular, $u_{k} \in H^{1} (Ω)$ for $k \geq 0$ , and $u_{k}^{-} \in H^{1} (Ω)$ for $k \leq 0$ . If $n \geq 2$ , $Ω$ is bounded and $C^{1}$ , and $u_{k} \in H_{0}^{1} (Ω)$ , then this membership is the boundary condition $u \leq k$ on $\partial Ω$ in the sense of Weak subsolutions and supersolutions of a divergence-form equation. For every $η \in C_{c}^{\infty} (Ω)$ the product $η^{2} u_{k}$ lies in $H_{0}^{1} (Ω)$ with $D (η^{2} u_{k}) = 2 η u_{k} D η + η^{2} D u_{k} a.e. on Ω,$ so $η^{2} u_{k}$ belongs to the Sobolev test space. It is admissible in the global $H^{- 1}$ formulation when the source pairing is continuous; for a local inequality with $f \in L_{l o c}^{2}$ , its pairing extends by density on a bounded neighborhood of the cutoff support. No pairing with a general $L_{l o c}^{1}$ source and an arbitrary $H_{0}^{1}$ test is asserted. The same membership conclusions hold for $k$ -translates of $u^{+}$ and for the cut-off functions $η^{2} u_{k}$ with $η \in W_{c}^{1, \infty} (Ω)$ .

Facts & Assumptions

Given: Countable Choice and the Axiom of Choice; an open set $Ω \subseteq R^{n}$ with $n \geq 1$ ; a real class $u \in H^{1} (Ω; R)$ ; a real level $k$ ; and $u_{k} = (u - k)^{+}$ , $u_{k}^{-} = (u - k)^{-}$ .

[F1]

$H^{1} (Ω; R) = W^{1, 2} (Ω; R)$ consists of the classes in $L^{2} (Ω; R)$ whose first weak derivatives exist as $L^{2}$ classes; the weak-derivative formula is the signed test identity (The notation $H^{k}$ and the reserved zero-boundary symbol, Integer-order Sobolev spaces and their norms).

[F2]

Assume the Axiom of Choice. For $u \in W^{1, p} (Ω; R)$ and $F : R \to R$ Lipschitz: $F \circ u \in W_{l o c}^{1, p} (Ω)$ with $D_{i} (F \circ u) = F^{'} (u) D_{i} u$ almost everywhere where $F$ is differentiable at $u$ , the product being taken as $0$ on the null level set $N_{F}$ ; moreover $F \circ u \in W^{1, p} (Ω)$ exactly when $F (u) \in L^{p} (Ω)$ (Chain rule for globally Lipschitz scalar maps of Sobolev functions).

[F3]

Assume the Axiom of Choice. For $w \in W^{1, p} (Ω; R)$ , $w^{+}, w^{-} \in W^{1, p} (Ω; R)$ with $D_{i} w^{+} = 1_{{w > 0}} D_{i} w$ , $D_{i} w^{-} = - 1_{{w < 0}} D_{i} w$ and $D_{i} w = 0$ almost everywhere on ${w = 0}$ (Positive, negative, and truncated Sobolev functions).

[F4]

Assume the Axiom of Choice. For $η \in C_{c}^{\infty} (Ω)$ and $w \in W^{1, p} (Ω)$ , the product $η w$ lies in $W^{1, p} (Ω)$ and $D_{i} (η w) = η D_{i} w + w D_{i} η$ almost everywhere (Weak Leibniz rule with a smooth factor, Bounded restriction and cutoff localisation in Sobolev spaces).

[F5]

Assume Countable Choice. If $w \in H^{1} (Ω)$ vanishes almost everywhere outside a compact set $K_{0} \subset Ω$ , then its zero extension lies in $H^{1} (R^{n})$ and is approximated in the $H^{1}$ norm by compactly supported smooth functions; choosing mollifier radii smaller than $dist (K_{0}, \partial Ω)$ and restricting the approximants exhibits $w$ as an $H^{1} (Ω)$ -limit of $C_{c}^{\infty} (Ω)$ functions, hence $w \in H_{0}^{1} (Ω)$ (Compactly supported Sobolev functions extend by zero in every integer order, Compactly supported smooth functions are dense in W^{k,p}(R^n), Zero-boundary Sobolev space as a norm closure).

[F6]

Weak boundary order: when $n \geq 2$ and $Ω$ is a bounded $C^{1}$ domain, $u \leq k$ on $\partial Ω$ means $(u - k)^{+} \in H_{0}^{1} (Ω)$ (Weak subsolutions and supersolutions of a divergence-form equation).

[F7]

Assume the Axiom of Choice. ACL product rule: if $η \in W_{c}^{1, \infty} (Ω; R)$ and $w \in H^{1} (Ω; R)$ , then $η w \in H^{1} (Ω; R)$ with $D_{i} (η w) = η D_{i} w + w D_{i} η$ almost everywhere. Indeed $η$ and $w$ have ACL representatives whose sections are absolutely continuous on almost every line (The ACL characterisation of $W^{1, p}$ ), the ordinary product rule holds along those lines, and the resulting a.e. line derivatives determine the weak derivative by the reconstruction lemma (ACL representatives recover their weak gradients by Fubini).

Proof

technique · direct; the truncations are produced by the globally Lipschitz chain rule, and the cut-off tests by the product rules and the compact-support characterisation of $H^1_0$

1.1givenF1F2

The function $F (t) : = (t - k)^{+}$ is Lipschitz with constant $1$ and differentiable off $k$ ; the chain rule [F2] gives $u_{k} \in H_{l o c}^{1} (Ω)$ and $D_{i} u_{k} = 1_{{u > k}} D_{i} u$ locally a.e. If $∣ Ω ∣ < \infty$ , the bound $u_{k} \leq ∣ u ∣ + ∣ k ∣$ gives $u_{k} \in L^{2} (Ω)$ and hence $H^{1} (Ω)$ ; if $k \geq 0$ , then $0 \leq u_{k} \leq u^{+}$ , which gives global membership without a finite-measure assumption. In general, global membership follows whenever $u_{k} \in L^{2} (Ω)$ , since its weak gradient is bounded by $∣ D u ∣$ .

1.2givenF1F2

Likewise $G (t) : = (t - k)^{-} = \max {k - t, 0}$ is Lipschitz with constant $1$ and differentiable off $k$ ; the chain rule gives $u_{k}^{-} \in H_{l o c}^{1} (Ω)$ and $D_{i} u_{k}^{-} = - 1_{{u < k}} D_{i} u$ locally a.e. If $∣ Ω ∣ < \infty$ , then $u_{k}^{-} \in L^{2} (Ω)$ and hence $H^{1} (Ω)$ ; if $k \leq 0$ , then $0 \leq u_{k}^{-} \leq u^{-}$ . In general, global membership follows whenever $u_{k}^{-} \in L^{2} (Ω)$ .

2.1step 1.1step 1.2F3

On ${u \leq k}$ the indicator $1_{{u > k}}$ vanishes, so the almost-everywhere identity of step 1.1 gives $D_{i} u_{k} = 0$ almost everywhere on ${u \leq k}$ , and a fortiori almost everywhere on ${u < k}$ ; at level $k = 0$ this is exactly the positive-part calculus of [F3] for $w = u$ , whose formula $D_{i} w^{+} = 1_{{w > 0}} D_{i} w$ agrees with step 1.1. The same argument applied to step 1.2 gives $D_{i} u_{k}^{-} = 0$ almost everywhere on ${u \geq k}$ .

2.2step 1.1F6given

If $n \geq 2$ and $Ω$ is a bounded $C^{1}$ domain, the equivalence " $u_{k} \in H_{0}^{1} (Ω)$ if and only if $u \leq k$ on $\partial Ω$ " is the definition of the weak boundary order in [F6], read with $u_{k} = (u - k)^{+}$ ; no pointwise boundary values are involved.

2.3step 1.1F4F5

Let $η \in C_{c}^{\infty} (Ω)$ and put $v : = η^{2} u_{k}$ . Choose a bounded neighborhood $V$ of $supp η$ with $\overline{V} \subset Ω$ . By step 1.1, $u_{k} \in H^{1} (V)$ ; the product rule [F4] gives $v \in H^{1} (Ω)$ with $D_{i} v = 2 η u_{k} D_{i} η + η^{2} D_{i} u_{k}$ almost everywhere. Its support is compact in $Ω$ , so [F5] gives $v \in H_{0}^{1} (Ω)$ . Since $η^{2} \geq 0$ and $u_{k} \geq 0$ , it is a nonnegative Sobolev test. If the source is in $L_{l o c}^{2}$ on $V$ , density extends the local inequality to this test; it is also valid for the global formulation whenever the source defines a continuous functional on $H_{0}^{1}$ . For a general $L_{l o c}^{1}$ source, membership alone does not assert that the pairing is defined.

3.1step 2.3F5F7

Now let $η \in W_{c}^{1, \infty} (Ω)$ . On a bounded neighborhood $V$ of its support, the ACL product rule [F7] applied twice gives $η^{2} u_{k} \in H^{1} (Ω)$ with $D_{i} (η^{2} u_{k}) = 2 η u_{k} D_{i} η + η^{2} D_{i} u_{k}$ almost everywhere. Its compact support and nonnegativity again give $η^{2} u_{k} \in H_{0}^{1} (Ω)$ ; admissibility in an inequality requires the same source-pairing condition as in step 2.3.

4.1step 1.1step 1.2step 2.3step 3.1F3∎

Apply steps 1.1-3.1 to $v = u^{+}$ and $v = u^{-}$ , both of which lie in $H^{1} (Ω; R)$ by [F3]. For every $κ \in R$ , each truncation $(v - κ)^{\pm}$ lies in $H_{l o c}^{1} (Ω)$ with the corresponding level-set gradient formula, and its cutoff products with $η \in C_{c}^{\infty} (Ω)$ or $η \in W_{c}^{1, \infty} (Ω)$ lie in $H_{0}^{1} (Ω)$ . Global $H^{1} (Ω)$ membership holds when $∣ Ω ∣ < \infty$ or when that truncation is in $L^{2} (Ω)$ ; in particular $(v - κ)^{+} \in H^{1} (Ω)$ for $κ \geq 0$ , while $(v - κ)^{-} = 0$ for $κ \leq 0$ because $v \geq 0$ . Admissibility in a weak inequality still requires the source pairing to extend continuously to the test space, as specified in the Definition. All steps use only Countable Choice and the Axiom of Choice as declared in [F2]-[F5] and [F7].

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

A function whose trace is at most a level has positive part in the zero-boundary space

Statement

Assume Countable Choice and the Axiom of Choice, inherited through the published trace and density suppliers named below. Let $n \geq 2$ , let $Ω \subset R^{n}$ be a bounded $C^{1}$ domain (Bounded C^k domains and boundary charts), let $u \in H^{1} (Ω; R)$ , and let $T$ be the trace operator of The $L^{p}$ trace operator on a bounded $C^{1}$ domain. Then $T u \leq 0$ a.e. on $\partial Ω$ implies $u^{+} \in H_{0}^{1} (Ω)$ , and conversely $u^{+} \in H_{0}^{1} (Ω)$ implies $T u \leq 0$ a.e.; more generally, for every $k \in R$ , $(u - k)^{+} \in H_{0}^{1} (Ω)$ if and only if $T u \leq k$ a.e. on $\partial Ω$ . In particular the weak boundary order of Weak subsolutions and supersolutions of a divergence-form equation is the pointwise trace order, and the two conventions give the same boundary supremum $\sup_{\partial Ω} u = {ess sup}_{\partial Ω} T u$ (with value $+ \infty$ only if the trace is not essentially bounded above).

Facts & Assumptions

Given: Countable Choice and the Axiom of Choice; a bounded $C^{1}$ domain $Ω \subset R^{n}$ , $n \geq 2$ ; a real class $u \in H^{1} (Ω; R)$ ; the trace $T$ ; and a real level $k$ .

[F1]

$T : W^{1, 2} (Ω; R) \to L^{2} (\partial Ω; R)$ is linear and bounded, and $T v = v ∣_{\partial Ω}$ for every $v \in C (\overline{Ω}) \cap H^{1} (Ω)$ (The $L^{p}$ trace operator on a bounded $C^{1}$ domain, The trace agrees with classical restriction for continuous Sobolev functions).

[F2]

Assume the Axiom of Choice. Smooth functions on $\overline{Ω}$ (restrictions of $C_{c}^{\infty} (R^{n})$ functions) are dense in $H^{1} (Ω)$ , and $C_{c}^{\infty} (Ω)$ is dense in $H_{0}^{1} (Ω)$ by definition (Ambient smooth restrictions are dense on bounded C^k domains, Zero-boundary Sobolev space as a norm closure).

[F3]

Assume the Axiom of Choice. ${w \in H^{1} (Ω) : T w = 0} = H_{0}^{1} (Ω)$ (The kernel of the trace is the closure of the test functions).

[F4]

Assume the Axiom of Choice. If $w, w_{j} \in H^{1} (Ω; R)$ with $w_{j} \to w$ in $H^{1}$ , then $w_{j}^{+} \to w^{+}$ in $H^{1}$ : pointwise $∣ w_{j}^{+} - w^{+} ∣ \leq ∣ w_{j} - w ∣$ and $D (w_{j}^{+} - w^{+}) = 1_{{w_{j} > 0}} D w_{j} - 1_{{w > 0}} D w = 1_{{w_{j} > 0}} D (w_{j} - w) + (1_{{w_{j} > 0}} - 1_{{w > 0}}) D w$ , whose first term tends to $0$ in $L^{2}$ . Every subsequence has a further subsequence with $w_{j} \to w$ a.e.: choose the further terms with $∥ w_{j} - w ∥_{2}^{2} \leq 2^{- 3 j}$ , so $∣ {∣ w_{j} - w ∣ > 2^{- j}} ∣ \leq 2^{- j}$ and countable subadditivity makes the limsup null. Along this further subsequence the indicator difference tends to zero where $w \neq 0$ , while $D w = 0$ a.e. where $w = 0$ ; dominated convergence with $4 ∣ D w ∣^{2}$ makes the second term tend to zero in $L^{2}$ . If the full positive-part sequence did not converge, a subsequence with errors bounded below would contradict this argument. Therefore $w_{j}^{+} \to w^{+}$ in $H^{1}$ (Positive-part truncation calculus and admissible cut-off weak tests, Positive, negative, and truncated Sobolev functions).

[F5]

Weak boundary order: $u \leq k$ on $\partial Ω$ means $(u - k)^{+} \in H_{0}^{1} (Ω)$ , and $\sup_{\partial Ω} u = \inf {k : u \leq k on \partial Ω}$ with $\inf \emptyset = + \infty$ (Weak subsolutions and supersolutions of a divergence-form equation).

Proof

technique · direct; approximate by functions smooth up to the boundary, where the trace is the boundary restriction and commutes with truncation, then pass to the limit

1.1givenF1F2

Fix $k \in R$ and put $w : = u - k \in H^{1} (Ω; R)$ and $w_{k} : = (u - k)^{+}$ . By [F2] choose $w_{j} \in C^{\infty} (\overline{Ω})$ with $w_{j} \to w$ in $H^{1}$ . For each $j$ , $w_{j}^{+}$ is continuous on $\overline{Ω}$ as the maximum of the continuous functions $w_{j}$ and $0$ , and it lies in $H^{1} (Ω)$ by the Lipschitz chain rule, so [F1] gives $T w_{j}^{+} = w_{j}^{+} ∣_{\partial Ω} = (w_{j} ∣_{\partial Ω})^{+} = (T w_{j})^{+}$ (the last equality using $T w_{j} = w_{j} ∣_{\partial Ω}$ from [F1]); moreover $T w_{j} \to T w$ in $L^{2} (\partial Ω)$ , so $(T w_{j})^{+} \to (T w)^{+}$ in $L^{2} (\partial Ω)$ because $t \mapsto t^{+}$ is $1$ -Lipschitz on $R$ .

2.1step 1.1F1F4

By [F4], $w_{j}^{+} \to w^{+}$ in $H^{1} (Ω)$ , so the continuity of $T$ in [F1] gives $T w_{j}^{+} \to T w^{+}$ in $L^{2} (\partial Ω)$ . Since step 1.1 gives $T w_{j}^{+} = (T w_{j})^{+} \to (T w)^{+}$ in the same space, uniqueness of $L^{2}$ limits yields $T (w^{+}) = (T w)^{+}$ a.e. on $\partial Ω$ .

3.1step 2.1F3algebra

Consequently, by [F3], $w^{+} \in H_{0}^{1} (Ω) ⟺ T w^{+} = 0$ in $L^{2} (\partial Ω) ⟺ (T w)^{+} = 0$ a.e. $⟺ T w \leq 0$ a.e. on $\partial Ω$ ; since $w = u - k$ and $w^{+} = (u - k)^{+}$ , this is the asserted equivalence $(u - k)^{+} \in H_{0}^{1} (Ω) ⟺ T u \leq k$ a.e.

4.1step 3.1F3F5∎

The boundary supremum. By [F5] and step 3.1, the admissible levels are ${k \in R : u \leq k$ on $\partial Ω} = {k \in R : T u \leq k$ a.e. $} = [{ess sup}_{\partial Ω} T u, + \infty)$ when the trace is essentially bounded above, and the empty set when it is not; the infimum is therefore ${ess sup}_{\partial Ω} T u$ in the first case and $+ \infty$ in the second, which proves the boundary-supremum identification. The argument uses only the declared Countable Choice and Axiom of Choice.

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Caccioppoli inequality for truncated subsolutions

Statement

Assume Countable Choice and the Axiom of Choice. Let $n \geq 2$ , let $Ω \subseteq R^{n}$ be open, and let $0 < θ \leq M_{a}^{2}$ , and let $A = (a^{i j})$ be measurable with $a^{i j} = a^{j i}$ and $θ ∣ ξ ∣^{2} \leq \sum_{i, j = 1}^{n} a^{i j} (x) ξ_{i} ξ_{j} \leq M_{a}^{2} ∣ ξ ∣^{2} for a.e. x \in Ω and all ξ \in R^{n} .$ Write $L_{0} u : = - D_{i} (a^{i j} D_{j} u)$ and $a_{0} (u, v) : = \int_{Ω} a^{i j} D_{j} u D_{i} v d x$ for real $u, v \in H^{1} (Ω)$ . Let $f \in L_{l o c}^{2} (Ω)$ and let $u \in H^{1} (Ω; R)$ satisfy the local weak subsolution inequality $a_{0} (u, φ) \leq \int_{Ω} f φ d x for every nonnegative φ \in C_{c}^{\infty} (Ω) .$ Then for every $k \in R$ and every $η \in C_{c}^{\infty} (Ω)$ with $0 \leq η \leq 1$ , $\int_{Ω} η^{2} ∣ D (u - k)^{+} ∣^{2} d x \leq \frac{4 M_{a}^{2}}{θ} \int_{Ω} (u - k)^{+ 2} ∣ D η ∣^{2} d x + \frac{2}{θ} \int_{Ω} η^{2} (u - k)^{+} f^{+} d x,$ and for concentric balls $B_{r} (x_{0}) ⋐ B_{R} (x_{0}) ⋐ Ω$ , $0 < r < R$ , $\int_{B_{r} (x_{0})} ∣ D (u - k)^{+} ∣^{2} d x \leq C (n, θ, M_{a}) (\frac{1}{(R - r)^{2}} \int_{B_{R} (x_{0})} (u - k)^{+ 2} d x + \int_{B_{R} (x_{0})} (u - k)^{+} f^{+} d x) .$ All integrands are restricted to the superlevel set ${u > k}$ , where $(u - k)^{+} > 0$ ; the estimate is uniform in $k$ and in the localisation.

Facts & Assumptions

Given: Countable Choice and the Axiom of Choice; an open $Ω \subseteq R^{n}$ , $n \geq 2$ ; constants $0 < θ \leq M_{a}^{2}$ ; a measurable symmetric coefficient field $A = (a^{i j})$ with $θ ∣ ξ ∣^{2} \leq ⟨ A ξ, ξ ⟩ \leq M_{a}^{2} ∣ ξ ∣^{2}$ a.e.; $f \in L_{l o c}^{2} (Ω)$ ; and a real class $u \in H^{1} (Ω; R)$ with $a_{0} (u, φ) \leq \int_{Ω} f φ$ for every nonnegative $φ \in C_{c}^{\infty} (Ω)$ .

[F1]

Assume Countable Choice and the Axiom of Choice. For $k \in R$ and $u \in H^{1} (Ω; R)$ , the class $u_{k} : = (u - k)^{+}$ lies in $H_{l o c}^{1} (Ω; R)$ with $D u_{k} = 1_{{u > k}} D u$ , and $D u_{k} = 0$ a.e. on ${u \leq k}$ . Global $H^{1} (Ω)$ membership is not asserted for arbitrary $k$ on an infinite-measure domain. For $η \in C_{c}^{\infty} (Ω)$ the product $η^{2} u_{k}$ lies in $H_{0}^{1} (Ω)$ , is nonnegative, and satisfies $D (η^{2} u_{k}) = 2 η u_{k} D η + η^{2} D u_{k}$ a.e. (Positive-part truncation calculus and admissible cut-off weak tests, Weak subsolutions and supersolutions of a divergence-form equation).

[F2]

Assume Countable Choice. Every element of $H^{1} (Ω)$ has weak first derivatives in $L^{2} (Ω)$ , the weak derivative is linear, and products of $L^{2}$ classes with bounded measurable coefficients are integrable on compact sets (Integer-order Sobolev spaces and their norms).

[F3]

Bumps: for $0 < r < R$ there is $η \in C_{c}^{\infty} (B_{R} (x_{0}))$ with $0 \leq η \leq 1$ , $η = 1$ on $B_{r} (x_{0})$ and $∣ D η ∣ \leq C_{U} / (R - r)$ for a universal constant $C_{U}$ ; the explicit radial bump $η (x) = σ ((s^{2} - ∣ x - x_{0} ∣^{2}) / (s^{2} - r^{2}))$ , $s = (r + R) / 2$ , of A smooth bump between concentric Euclidean balls and Compactly supported scaled Euclidean bumps provides it, since on the support $∣ x - x_{0} ∣ \leq s$ and the chain rule give $∣ D η ∣ \leq ∥ σ^{'} ∥_{\infty} 2 s / (s^{2} - r^{2}) \leq ∥ σ^{'} ∥_{\infty} 4 / (R - r)$ .

[F4]

Young's inequality with conjugate exponents $p = q = 2$ and weight: for $a, b \geq 0$ and $ε > 0$ , $2 a b \leq ε a^{2} + b^{2} / ε$ (Young's inequality for conjugate real exponents); Cauchy–Schwarz in $L^{2}$ gives $∣ \int g h ∣ \leq (\int g^{2})^{1 / 2} (\int h^{2})^{1 / 2}$ (Holder's inequality for integrals, including the endpoint cases).

Proof

technique · direct; insert the truncated test function into the subsolution inequality, expand, and absorb the cross term by Young's inequality

1.1givenF1F2F4algebra

Fix $k$ and $η \in C_{c}^{\infty} (Ω)$ with $0 \leq η \leq 1$ , and put $u_{k} : = (u - k)^{+}$ and $v : = η^{2} u_{k}$ . By [F1], $v \in H_{0}^{1} (Ω)$ is nonnegative; since $f \in L_{l o c}^{2}$ and the support is compact, density extends the local subsolution inequality to this test. Thus $a_{0} (u, v) \leq \int_{Ω} η^{2} u_{k} f^{+} d x$ . Expanding and using $D u_{k} = 1_{{u > k}} D u$ , define $S^{2} : = \int_{Ω} η^{2} a^{i j} D_{j} u_{k} D_{i} u_{k} d x$ . The correct identity is $S^{2} = a_{0} (u, v) - 2 \int_{Ω} η u_{k} a^{i j} D_{j} u_{k} D_{i} η d x \leq \int_{Ω} η^{2} u_{k} f^{+} d x + 2 ∣ \int_{Ω} η u_{k} a^{i j} D_{j} u_{k} D_{i} η d x ∣ .$ The matrix Cauchy--Schwarz inequality and $A \leq M_{a}^{2} I$ bound the last term by $2 ∣ M_{a} ∣ S (\int_{Ω} u_{k}^{2} ∣ D η ∣^{2} d x)^{1 / 2}$ .

2.1step 1.1F4algebra

By Young's inequality $2 ∣ M_{a} ∣ S b \leq \frac{1}{2} S^{2} + 2 M_{a}^{2} b^{2}$ with $b = (\int_{Ω} u_{k}^{2} ∣ D η ∣^{2})^{1 / 2}$ , step 1.1 gives $S^{2} \leq 2 \int_{Ω} η^{2} u_{k} f^{+} d x + 4 M_{a}^{2} \int_{Ω} u_{k}^{2} ∣ D η ∣^{2} d x$ . Ellipticity gives $S^{2} \geq θ \int_{Ω} η^{2} ∣ D u_{k} ∣^{2} d x$ , and therefore $\int_{Ω} η^{2} ∣ D u_{k} ∣^{2} d x \leq \frac{4 M_{a}^{2}}{θ} \int_{Ω} u_{k}^{2} ∣ D η ∣^{2} d x + \frac{2}{θ} \int_{Ω} η^{2} u_{k} f^{+} d x .$ This is the first estimate.

3.1step 2.1F3∎

For the ball form let $0 < r < R$ with $B_{R} (x_{0}) ⋐ Ω$ and choose the bump $η$ of [F3], so that $0 \leq η \leq 1$ , $η = 1$ on $B_{r} (x_{0})$ , $supp η \subseteq B_{R} (x_{0})$ and $∣ D η ∣ \leq C_{U} / (R - r)$ . Applying step 2.1 gives the second estimate, with $C (n, θ, M_{a}) : = \max {4 M_{a}^{2} C_{U}^{2} / θ, 2 / θ}$ . Both estimates are uniform in $k$ , and no choice principle beyond the declared Countable Choice and Axiom of Choice is used.

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Sobolev level-set step: energy decay with explicit level gap and radius loss

Statement

Assume Countable Choice and the Axiom of Choice. Let $n \geq 3$ , let $B_{R} \subset R^{n}$ be a ball, and let $u \in H^{1} (B_{R}; R)$ . Suppose there is $C_{0} \geq 1$ such that for every $0 < ρ < R$ and every level $k$ $\int_{B_{ρ}} ∣ D (u - k)^{+} ∣^{2} d x \leq C_{0} (R - ρ)^{- 2} \int_{B_{R}} (u - k)^{+ 2} d x,$ i.e. the truncated Caccioppoli estimate of Caccioppoli inequality for truncated subsolutions holds with $f = 0$ on $B_{R}$ . Then there is $C = C (n, C_{0})$ such that for all $0 < r < R$ and all $h < k$ : $\int_{B_{r}} (u - k)^{+ 2} d x \leq C (R - r)^{- 2} (k - h)^{- 4 / n} (\int_{B_{R}} (u - h)^{+ 2} d x)^{1 + 2 / n},$ and consequently, for $u \geq 0$ and $h > 0$ , $∣ {u > k} \cap B_{r} ∣ \leq C (k - h)^{- 2} (R - r)^{- 2} h^{- 4 / n} (\int_{B_{R}} u^{2} d x)^{1 + 2 / n} .$ For $n = 2$ and each $0 < δ < 1$ , the same conclusions hold with $1 + δ$ in place of $1 + 2 / n$ , $(k - h)^{- 2 δ}$ and $h^{- 2 δ}$ in place of the powers $- 4 / n$ , and the common factor $(R - r)^{- 2}$ replaced by $R^{2 - 2 δ} (R - r)^{- 2}$ . Here the constant may also depend on $δ$ . Indeed, the critical Sobolev inequality on $B_{R}$ has the scaled form $∥ v ∥_{L^{κ} (B_{R})} \leq S_{κ} R^{2 / κ} ∥ D v ∥_{L^{2} (B_{R})}$ for finite $κ > 2$ , and choosing $κ = 2 / (1 - δ)$ gives $δ = 1 - 2 / κ$ . Thus the open range $0 < δ < 1$ is exactly the range supplied by finite $κ$ , and the radius factor is the one dictated by dilation (The critical Sobolev embedding into every finite $L^{q}$ , The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction).

Facts & Assumptions

Given: Countable Choice and the Axiom of Choice; $n \geq 2$ ; a ball $B_{R}$ ; a real class $u \in H^{1} (B_{R}; R)$ ; a constant $C_{0} \geq 1$ with $\int_{B_{ρ}} ∣ D (u - k)^{+} ∣^{2} \leq C_{0} (R - ρ)^{- 2} \int_{B_{R}} (u - k)^{+ 2}$ for every $0 < ρ < R$ and every level $k$ ; radii $0 < r < R$ and levels $h < k$ .

[F1]

Assume Countable Choice. For $k \in R$ the class $u_{k} = (u - k)^{+}$ lies in $H^{1} (B_{R}; R)$ , and for $η \in C_{c}^{\infty} (B_{R})$ the product $η u_{k}$ lies in $H_{0}^{1} (B_{R})$ with $D (η u_{k}) = η D u_{k} + u_{k} D η$ almost everywhere (Positive-part truncation calculus and admissible cut-off weak tests, Integer-order Sobolev spaces and their norms).

[F2]

Assume the Axiom of Choice. Sobolev inequality: for $n \geq 3$ there is $S = S (n) < \infty$ with $∥ v ∥_{L^{2^{*}} (B_{R})} \leq S ∥ D v ∥_{L^{2} (B_{R})}$ for every $v \in H_{0}^{1} (B_{R})$ , where $2^{*} = 2 n / (n - 2)$ (The Sobolev inequality for zero-boundary Sobolev closures on open sets, The Sobolev conjugate exponent and the scaling identity).

[F3]

Assume the Axiom of Choice. On the unit ball in $R^{2}$ , the critical embedding into every finite $L^{κ}$ , combined with Poincaré's inequality for $H_{0}^{1}$ , gives $∥ v ∥_{L^{κ} (B_{1})} \leq S_{κ} ∥ D v ∥_{L^{2} (B_{1})}$ for finite $κ > 2$ . Dilation therefore gives $∥ v ∥_{L^{κ} (B_{R})} \leq S_{κ} R^{2 / κ} ∥ D v ∥_{L^{2} (B_{R})}$ for $v \in H_{0}^{1} (B_{R})$ (The critical Sobolev embedding into every finite $L^{q}$ , The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction).

[F4]

Chebyshev's inequality: for a nonnegative measurable $v$ and $t > 0$ , $∣ {v > t} ∣ \leq t^{- 2} \int v^{2}$ ; and Hölder's inequality gives $\int_{E} f^{2} \leq ∣ E ∣^{2 / n} ∥ f ∥_{L^{2^{*}}}^{2}$ for measurable $E$ of finite measure (Chebyshev-Markov inequality for the integral, Holder's inequality for integrals, including the endpoint cases, The space $L^{p} (μ)$ as the quotient by null functions).

[F5]

The radial cutoff used in the ball-form Caccioppoli estimate has a universal gradient constant: for $0 < r < ρ$ , take $s = (r + ρ) / 2$ and $η (x) = σ ((s^{2} - ∣ x ∣^{2}) / (s^{2} - r^{2}))$ . Then $η \in C_{c}^{\infty} (B_{ρ})$ , $0 \leq η \leq 1$ , $η = 1$ on $B_{r}$ , and $∣ D η ∣ \leq C_{U} / (ρ - r)$ with $C_{U} = 4 ∥ σ^{'} ∥_{\infty}$ , since on the support $∣ x ∣ \leq s$ and $s - r = (ρ - r) / 2$ ; this is the explicit cutoff calculation in Caccioppoli inequality for truncated subsolutions.

Proof

technique · direct; combine the truncated Caccioppoli estimate with the Sobolev embedding and Chebyshev's inequality, then iterate the resulting measure–energy inequality once

1.1givenF4

Put $w : = (u - k)^{+}$ and $v : = (u - h)^{+}$ . Since $h < k$ one has $0 \leq w \leq v$ , and ${w > 0} = {u > k} = {v > k - h} \subseteq {v > 0}$ . Applying Chebyshev's inequality to the nonnegative function $v$ at level $k - h > 0$ gives $∣ {w > 0} \cap B_{r} ∣ \leq ∣ {v > k - h} ∣ \leq (k - h)^{- 2} \int_{B_{R}} v^{2} d x$ .

1.2givenF1F5algebra

Choose $ρ : = (R + r) / 2 \in (r, R)$ and the bump $η$ of [F5] with $0 \leq η \leq 1$ , $η = 1$ on $B_{r}$ , $supp η \subseteq B_{ρ}$ and $∣ D η ∣ \leq C_{U} / (ρ - r) = 2 C_{U} / (R - r)$ . By [F1], $η w \in H_{0}^{1} (B_{R})$ , and $D (η w) = η D w + w D η$ almost everywhere, so the Caccioppoli hypothesis at radius $ρ$ and level $k$ , together with the elementary bound $(a + b)^{2} \leq 2 a^{2} + 2 b^{2}$ , gives $\int_{B_{R}} ∣ D (η w) ∣^{2} d x \leq 2 \int_{B_{ρ}} η^{2} ∣ D w ∣^{2} d x + 2 \int_{B_{ρ}} w^{2} ∣ D η ∣^{2} d x \leq \frac{8 (C_{0} + C_{U}^{2})}{(R - r)^{2}} \int_{B_{R}} w^{2} d x .$

2.1step 1.1F2F4algebra

Assume $n \geq 3$ and let $2^{*} = 2 n / (n - 2)$ . Applying the Sobolev inequality [F2] to $η w \in H_{0}^{1} (B_{R})$ and then Hölder's inequality [F4] on the support of $η w$ , which is contained in ${w > 0} \cap B_{ρ}$ up to a null set, gives $\int_{B_{r}} w^{2} d x \leq \int_{B_{R}} (η w)^{2} d x \leq ∣ {w > 0} \cap B_{ρ} ∣^{2 / n} S^{2} \int_{B_{R}} ∣ D (η w) ∣^{2} d x$ .

3.1step 1.1step 1.2step 2.1algebra

Substituting the bound of step 1.2 into step 2.1 and then the Chebyshev bound of step 1.1, and using $\int_{B_{R}} w^{2} \leq \int_{B_{R}} v^{2}$ , yields $\int_{B_{r}} (u - k)^{+ 2} d x \leq C (n, C_{0}) (R - r)^{- 2} (k - h)^{- 4 / n} (\int_{B_{R}} (u - h)^{+ 2} d x)^{1 + 2 / n}$ , which is the first displayed estimate.

3.2step 1.1step 1.2step 2.1F3F4algebra

Assume $n = 2$ and fix a finite exponent $κ > 2$ , put $δ : = 1 - 2 / κ \in (0, 1)$ , and fix $η$ and $w$ as in steps 1.1-1.2. Replacing $2^{*}$ by $κ$ in step 2.1, using the scaled inequality [F3] and Hölder in the form $\int_{E} f^{2} \leq ∣ E ∣^{1 - 2 / κ} ∥ f ∥_{L^{κ}}^{2}$ , and inserting steps 1.1 and 1.2 gives $\int_{B_{r}} (u - k)^{+ 2} d x \leq C (κ, C_{0}) R^{2 - 2 δ} (R - r)^{- 2} (k - h)^{- 2 δ} (\int_{B_{R}} (u - h)^{+ 2} d x)^{1 + δ}$ . As finite $κ > 2$ varies, $δ = 1 - 2 / κ$ ranges over exactly $(0, 1)$ ; the factor $R^{2 - 2 δ}$ is precisely the dilation factor from [F3].

4.1step 3.1step 3.2F4algebra

For the measure clause assume $u \geq 0$ and $h > 0$ . On ${u > k} \cap B_{r}$ one has $(u - h)^{+} > k - h > 0$ , so Chebyshev's inequality gives $∣ {u > k} \cap B_{r} ∣ \leq (k - h)^{- 2} \int_{B_{r}} (u - h)^{+ 2} d x$ , and the first estimate applied with levels $0 < h$ bounds the integral by the displayed energy expression, since $u \geq 0$ ; multiplying the two bounds gives the displayed measure estimate. For $n = 2$ the same argument carries the factor $R^{2 - 2 δ} (R - r)^{- 2} h^{- 2 δ}$ from step 3.2.

5.1step 3.1step 3.2step 4.1∎

Both displayed estimates follow from steps 3.1-4.1 with constants depending only on $n$ , the Sobolev constants and $C_{0}$ ; the hypothesis list uses the Caccioppoli estimate of Caccioppoli inequality for truncated subsolutions and the declared Countable Choice and Axiom of Choice only, so no further choice principle is used.

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The nonlinear geometric iteration: an explicit threshold forces convergence to zero

Statement

Let $δ > 0$ , $C \geq 1$ , $B \geq 1$ , and let $(Y_{j})_{j \geq 0}$ be a sequence of nonnegative real numbers with $Y_{j + 1} \leq C B^{j} Y_{j}^{1 + δ} (j \geq 0) .$ Put $λ : = (2 B)^{- 1 / δ} \in (0, 1)$ . If $Y_{0} \leq C^{- 1 / δ} (2 B)^{- 1 / δ^{2}}$ , then $Y_{j} \leq Y_{0} λ^{j} (j \geq 0),$ so in particular $Y_{j} \to 0$ and $\sum_{j \geq 0} Y_{j} < + \infty$ . Equivalently: the explicit smallness condition $C Y_{0}^{δ} \leq (2 B)^{- 1 / δ}$ on the initial datum forces geometric decay of the whole sequence with ratio $λ$ .

Facts & Assumptions

Given: real numbers $δ > 0$ , $C \geq 1$ , $B \geq 1$ , and a sequence $(Y_{j})_{j \geq 0}$ of nonnegative reals with $Y_{j + 1} \leq C B^{j} Y_{j}^{1 + δ}$ for all $j \geq 0$ ; put $λ = (2 B)^{- 1 / δ}$ .

[F1]

Real powers with positive base: $λ^{δ} = 1 / (2 B)$ , so $B λ^{δ} = 1 / 2$ ; moreover $0 < λ < 1$ because $2 B \geq 2$ , and for $a > 0$ and real $u, v$ one has $a^{u + v} = a^{u} a^{v}$ , $(a^{u})^{v} = a^{u v}$ and $a^{0} = 1$ (Real powers for positive bases, with the zero-base positive-exponent convention). Also $t \mapsto t^{1 + δ}$ is nondecreasing on $[0, + \infty)$ because $1 + δ > 0$ .

[L1]

For $0 < λ < 1$ the sequence of integer powers $(λ^{j})_{j \geq 0}$ is null (For $∣ r ∣ < 1$ the sequence $r^{k}$ is null, and for $∣ r ∣ > 1$ the sequence $∣ r ∣^{k}$ diverges to $+ \infty$ ).

Proof

technique · direct induction with the ansatz $Y_j\le Y_0\lambda^j$, using that the induction requirement is largest at $j=0$

1.1givenF1algebra

The hypothesis is equivalent to $C Y_{0}^{δ} \leq λ$ : raising $Y_{0} \leq C^{- 1 / δ} (2 B)^{- 1 / δ^{2}}$ to the power $δ > 0$ gives $Y_{0}^{δ} \leq C^{- 1} (2 B)^{- 1 / δ} = C^{- 1} λ$ , and conversely this inequality implies the original one by raising to the power $1 / δ > 0$ and using the power identities of [F1]. Together with [F1] the data therefore satisfy $0 < λ < 1$ , $B λ^{δ} = 1 / 2$ and $C Y_{0}^{δ} \leq λ$ .

2.1step 1.1F1givenalgebra

Induction claim: $Y_{j} \leq Y_{0} λ^{j}$ for every $j \geq 0$ . The case $j = 0$ is $Y_{0} \leq Y_{0}$ . Assume the claim for some $j \geq 0$ . Then the recursion, the nonnegativity of $Y_{j}$ and the induction hypothesis give $Y_{j + 1} \leq C B^{j} Y_{j}^{1 + δ} \leq C B^{j} (Y_{0} λ^{j})^{1 + δ} = C B^{j} Y_{0}^{1 + δ} λ^{j (1 + δ)}$ , so it suffices to show $C B^{j} Y_{0}^{δ} λ^{j δ - 1} \leq 1$ , equivalently $Y_{0} λ^{j + 1} \geq C B^{j} Y_{0}^{1 + δ} λ^{j (1 + δ)}$ . By step 1.1 and [F1], $C B^{j} Y_{0}^{δ} λ^{j δ - 1} = (C Y_{0}^{δ} / λ) (B λ^{δ})^{j} = (C Y_{0}^{δ} / λ) 2^{- j} \leq C Y_{0}^{δ} / λ \leq 1$ . Hence $Y_{j + 1} \leq Y_{0} λ^{j + 1}$ , and induction proves the claim for all $j$ .

3.1step 2.1L1F1algebra∎

By step 2.1, $0 \leq Y_{j} \leq Y_{0} λ^{j}$ with $0 < λ < 1$ , so $Y_{j} \to 0$ by [L1]. Moreover the finite geometric sum identity $(1 - λ) \sum_{j = 0}^{N} λ^{j} = 1 - λ^{N + 1}$ , proved by induction on $N$ , gives $\sum_{j = 0}^{N} Y_{j} \leq Y_{0} \sum_{j = 0}^{N} λ^{j} \leq Y_{0} / (1 - λ)$ for every $N$ , because $λ^{N + 1} \geq 0$ ; the partial sums of the nonnegative series $\sum_{j \geq 0} Y_{j}$ are therefore increasing and bounded above by $Y_{0} / (1 - λ)$ , so the series converges and $\sum_{j \geq 0} Y_{j} \leq Y_{0} / (1 - λ) < + \infty$ . Only the displayed power identities and the null geometric sequence are used, so no choice principle is used.

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Weak maximum principle for coercive divergence-form equations

Statement

Assume Countable Choice and the Axiom of Choice through the Poincare and Sobolev suppliers below. Let $n \geq 2$ , let $Ω \subset R^{n}$ be a bounded $C^{1}$ domain, and let $L$ , $a$ and the real coefficient functions $a^{i j}, b^{i}, c$ be as in Weak subsolutions and supersolutions of a divergence-form equation, with ellipticity constant $θ$ and bounds $M_{a}, M_{b}, M_{c}$ . Let $f \in L_{l o c}^{1} (Ω; R)$ and $u \in H^{1} (Ω; R)$ be a real local weak subsolution of $L u = f$ . Suppose the weak sign condition $\int_{Ω} (c ζ + b^{i} D_{i} ζ) d x \geq 0 for every ζ \in C_{c}^{\infty} (Ω), ζ \geq 0,$ holds, and assume $c \geq 0$ a.e. on $Ω$ . Then:

Homogeneous case. If $f = 0$ a.e., then ${ess sup}_{Ω} u \leq \sup_{\partial Ω} u^{+},$ with the boundary supremum of Weak subsolutions and supersolutions of a divergence-form equation. If in addition $b \equiv 0$ , $c \equiv 0$ , and $u$ is a weak solution of $L u = 0$ , then ${ess sup}_{Ω} u = \sup_{\partial Ω} u$ .
Forcing with signed lower order. If $b \equiv 0$ , $c \geq 0$ a.e. and $f \in L^{q} (Ω)$ for some $q > n / 2$ ( $q > 1$ when $n = 2$ ), then the local inequality extends to all nonnegative $H_{0}^{1} (Ω)$ tests and ${ess sup}_{Ω} u \leq \sup_{\partial Ω} u^{+} + C ∥ f^{+} ∥_{L^{q} (Ω)}, C = C (n, q, θ, M_{a}, M_{c}, Ω),$ where $Ω$ enters $C$ only through its Poincare constant and volume.

If $u$ is a weak supersolution of $L u = f$ under either set of hypotheses, apply the corresponding bound to $- u$ for the same operator coefficients $(a, b, c)$ and source $- f$ . This gives ${ess inf}_{Ω} u \geq - \sup_{\partial Ω} u^{-}$ in the homogeneous case and ${ess inf}_{Ω} u \geq - \sup_{\partial Ω} u^{-} - C ∥ f^{-} ∥_{L^{q} (Ω)}$ in the forcing case. The maximum-principle conclusions concern real-valued classes and real coefficients.

Facts & Assumptions

Given: Countable Choice and the Axiom of Choice; a bounded $C^{1}$ domain $Ω \subset R^{n}$ , $n \geq 2$ ; real coefficients $a^{i j}, b^{i}, c \in L^{\infty} (Ω)$ with $θ ∣ ξ ∣^{2} \leq ⟨ A ξ, ξ ⟩$ and $∣ a^{i j} ∣ \leq M_{a}$ , $∣ b^{i} ∣ \leq M_{b}$ , $∣ c ∣ \leq M_{c}$ a.e.; $f \in L^{q} (Ω)$ with $q > n / 2$ ; and a weak subsolution $u \in H^{1} (Ω; R)$ satisfying the weak sign condition.

[F1]

Assume the Axiom of Choice. Trace, boundary order and truncation: $\sup_{\partial Ω} u = {ess sup}_{\partial Ω} T u$ and $(u - k)^{+} \in H_{0}^{1} (Ω)$ if and only if $T u \leq k$ a.e.; moreover $(u - k)^{+} \in H^{1} (Ω)$ with $D (u - k)^{+} = 1_{{u > k}} D u$ , and for $η \in C_{c}^{\infty} (Ω)$ the class $η^{2} (u - k)^{+}$ is an admissible nonnegative test (A function whose trace is at most a level has positive part in the zero-boundary space, Positive-part truncation calculus and admissible cut-off weak tests, Weak subsolutions and supersolutions of a divergence-form equation, The $L^{p}$ trace operator on a bounded $C^{1}$ domain, The kernel of the trace is the closure of the test functions).

[F2]

Sobolev inputs, all in the stated dimension $n \geq 2$ . The Gagliardo--Nirenberg--Sobolev inequality is stated for $C_{c}^{\infty} (R^{n})$ (The p=1 Gagliardo-Nirenberg-Sobolev inequality); if $w \in W_{0}^{1, 1} (Ω)$ , approximate it in $W^{1, 1}$ by $C_{c}^{\infty} (Ω)$ , extend each approximant by zero to $R^{n}$ , and pass to the limit to get $∥ w ∥_{L^{n / (n - 1)} (Ω)} \leq C (n) ∥ D w ∥_{L^{1} (Ω)}$ . Holder on measurable $E \subseteq Ω$ then gives $\int_{E} ∣ w ∣ d x \leq C (n) ∣ E ∣^{1 / n} \int_{Ω} ∣ D w ∣ d x$ . The density and zero-extension convention is Zero-boundary Sobolev space as a norm closure, and the Sobolev norms are those of Integer-order Sobolev spaces and their norms. For $n \geq 3$ and $1 \leq p < n$ there is $S = S (n, p)$ with $∥ w ∥_{L^{p^{*}}} \leq S ∥ D w ∥_{L^{p}}$ for $w \in W_{0}^{1, p} (Ω)$ , $p^{*} = n p / (n - p)$ (The Sobolev inequality for zero-boundary Sobolev closures on open sets); for $n = 2$ the embedding $W^{1, 2} (Ω) ↪ L^{κ} (Ω)$ holds on bounded extension domains for every finite $κ \geq 1$ (The critical Sobolev embedding into every finite $L^{q}$ ). In particular, on the bounded $C^{1}$ domain $Ω$ , for $n \geq 3$ one has $H_{0}^{1} (Ω) ↪ L^{κ} (Ω)$ for $2 \leq κ \leq 2^{*}$ , while for $n = 2$ every finite $κ \geq 2$ is available, with corresponding constants $S_{κ}$ . In dimension two these zero-boundary constants require only the volume: for $κ > 2$ , set $p = 2 κ / (κ + 2) \in (1, 2)$ , so $p^{*} = κ$ . Finite measure makes $H_{0}^{1} (Ω) \subset W_{0}^{1, p} (Ω)$ by the same smooth approximants, and the zero-boundary Sobolev inequality gives $∥ w ∥_{κ} \leq C (2, p) ∥ D w ∥_{p} \leq C (2, p) ∣ Ω ∣^{1 / κ} ∥ D w ∥_{2}$ . This proves the claimed dependence of the forcing constant on volume and Poincare constant alone.

[F3]

Poincare inequality on $W_{0}^{1, 2} (Ω)$ : there is $C_{P} = C_{P} (Ω)$ with $∥ w ∥_{L^{2} (Ω)} \leq C_{P} ∥ D w ∥_{L^{2} (Ω)}$ for every $w \in H_{0}^{1} (Ω)$ (The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction).

[F4]

Chebyshev and Holder: $∣ {w > t} ∣ \leq t^{- p} \int w^{p}$ for nonnegative measurable $w$ ; and for exponents $1 \leq r < κ$ one has $∥ w ∥_{L^{r} (E)} \leq ∣ E ∣^{1 / r - 1 / κ} ∥ w ∥_{L^{κ} (E)}$ for measurable $E$ of finite measure (Chebyshev-Markov inequality for the integral, Holder's inequality for integrals, including the endpoint cases, The space $L^{p} (μ)$ as the quotient by null functions, The essential supremum of a measurable function with respect to a measure, The essential supremum is attained as the least essential bound).

[F5]

Nonlinear iteration: if $Y_{j + 1} \leq C B^{j} Y_{j}^{1 + δ}$ with $C, B \geq 1$ , $δ > 0$ and $Y_{0} \leq C^{- 1 / δ} (2 B)^{- 1 / δ^{2}}$ , then $Y_{j} \leq Y_{0} λ^{j}$ with $λ = (2 B)^{- 1 / δ}$ and $Y_{j} \to 0$ (The nonlinear geometric iteration: an explicit threshold forces convergence to zero).

Proof

technique · use the weak sign condition on the first- and second-order truncation tests for the homogeneous maximum bound; for forcing, combine the energy estimate with a De Giorgi iteration whose finite Sobolev exponent is chosen to make the recurrence superlinear

1.1givenF1F3algebra

Homogeneous maximum bound. Put $t : = \sup_{\partial Ω} u^{+}$ . If $t = + \infty$ the bound is immediate. Otherwise $t \geq 0$ and $w : = (u - t)^{+} \in H_{0}^{1} (Ω)$ by [F1]. Since the equation is homogeneous, boundedness of the form and density extend its inequality from nonnegative compactly supported smooth tests to all nonnegative $H_{0}^{1}$ tests. For the weak sign condition, choose real $ϕ_{j} \in C_{c}^{\infty} (Ω)$ with $ϕ_{j} \to w$ in $H_{0}^{1}$ . Then $ϕ_{j}^{2} \geq 0$ and $ϕ_{j}^{2} \to w^{2}$ in $W^{1, 1}$ , since Cauchy--Schwarz gives convergence of both the functions and their gradients. Thus $w^{2} \in W_{0}^{1, 1}$ with $D (w^{2}) = 2 w D w$ , and boundedness of $b, c$ makes $ζ \mapsto \int (c ζ + b^{i} D_{i} ζ)$ continuous on $W^{1, 1}$ ; the sign condition therefore holds on $w^{2}$ without asserting $w^{2} \in H_{0}^{1}$ . Testing with $w$ and using $u = w + t$ on ${w > 0}$ gives $0 \geq a (u, w) = \int_{Ω} a^{i j} D_{j} w D_{i} w + \int_{Ω} (c w^{2} + b^{i} D_{i} w w) + t \int_{Ω} c w .$ The lower-order quadratic term is $\int_{Ω} (c w^{2} + b^{i} D_{i} w w) = \frac{1}{2} \int_{Ω} c w^{2} + \frac{1}{2} \int_{Ω} (c w^{2} + b^{i} D_{i} (w^{2})) \geq 0,$ by $c \geq 0$ and the extended weak sign condition; the boundary-shift term $t \int c w$ is nonnegative as well. Hence $θ ∥ D w ∥_{2}^{2} \leq 0$ , and Poincare gives $w = 0$ . Therefore ${ess sup}_{Ω} u \leq t$ .

1.2givenF2F3F4algebra

Forcing energy bound. Assume $b \equiv 0$ , $c \geq 0$ , and $f \in L^{q} (Ω)$ with the stated exponent. If $t : = \sup_{\partial Ω} u^{+} = + \infty$ , the claim is immediate; otherwise set $v : = (u - t)^{+} \in H_{0}^{1} (Ω)$ . Since $q > n / 2$ , Sobolev and Holder show that $f$ defines a continuous functional on $H_{0}^{1} (Ω)$ , so the local subsolution inequality extends to this test. On ${v > 0}$ , $u = v + t$ , and testing gives $θ ∥ D v ∥_{2}^{2} \leq \int_{Ω} f^{+} v \leq ∥ f^{+} ∥_{q} ∥ v ∥_{q^{'}} .$ For $n \geq 3$ take $κ = 2^{*}$ ; for $n = 2$ take any finite $κ > q^{'}$ . Holder, Poincare and the available Sobolev embedding imply $∥ v ∥_{q^{'}} \leq C ∥ D v ∥_{2}$ . Thus $∥ v ∥_{H_{0}^{1}} + ∥ v ∥_{2} \leq C_{E} ∥ f^{+} ∥_{q}$ , where constants depend only on the parameters in the Statement.

2.1step 1.2F2F3F4F5algebra

The forcing iteration. Write $a : = ∥ f^{+} ∥_{L^{q} (Ω)}$ . If $a = 0$ , step 1.2 gives $v = 0$ . Otherwise fix $T > 0$ and define $k_{j} = t + T (1 - 2^{- j})$ , $E_{j} : = {u > k_{j}}$ , $Y_{j} : = \int_{E_{j}} (u - k_{j})^{2}$ , and $B_{j} : = \int_{Ω} ∣ D (u - k_{j})^{+} ∣^{2}$ . Let $q^{'}$ be conjugate to $q$ , and choose $κ = 2^{*}$ for $n \geq 3$ ; for $n = 2$ choose finite $κ > 2 q^{'}$ . Set $δ : = 1 - 2 / κ > 0$ , $γ : = 1 / q^{'} - 1 / κ > 0$ , and $β : = δ + 2 γ$ . For $n \geq 3$ , $δ = 2 / n$ and $q > n / 2$ gives $β = 1 + 4 / n - 2 / q > 1$ ; for $n = 2$ , $β = 1 + 2 / q^{'} - 4 / κ > 1$ by the choice of $κ$ . Testing with $(u - k_{j})^{+}$ and using $c \geq 0$ , Holder on $E_{j}$ , and Sobolev gives $B_{j}^{1 / 2} \leq C a ∣ E_{j} ∣^{γ}$ (if $B_{j} = 0$ , Poincare gives $(u - k_{j})^{+} = 0$ ). Also $∣ E_{j + 1} ∣ \leq (T 2^{- j - 1})^{- 2} Y_{j}$ and Sobolev gives $Y_{j + 1} \leq C ∣ E_{j + 1} ∣^{δ} B_{j + 1}$ . Consequently $Y_{j + 1} \leq C_{0} a^{2} T^{- 2 β} 2^{2 β (j + 1)} Y_{j}^{β} .$ Set $B : = 2^{2 β}$ and $Z_{j} : = Y_{j} / T^{2}$ . Choose $T = C_{1} a$ with $C_{1}^{2} \geq C_{0} B$ and $C_{1}^{2} \geq C_{E} (2 B)^{1 / (β - 1)^{2}}$ , where $Y_{0} \leq C_{E} a^{2}$ by step 1.2. Then $Z_{j + 1} \leq (C_{0} B / C_{1}^{2}) B^{j} Z_{j}^{β} \leq B^{j} Z_{j}^{1 + (β - 1)}$ and $Z_{0} \leq (2 B)^{- 1 / (β - 1)^{2}}$ . The nonlinear iteration [F5] gives $Z_{j} \to 0$ , hence $Y_{j} \to 0$ . Since $(u - t - T)^{+} \leq (u - k_{j})^{+}$ and $(u - t - T)^{+} \in L^{2} (Ω)$ , this forces $(u - t - T)^{+} = 0$ a.e. on $Ω$ , proving the forcing bound. The finite $κ$ choice in dimension two uses the full open range of the critical Sobolev embedding.

3.1step 1.1F1F3algebra∎

Supersolutions and equality. If $u$ is a weak supersolution of $L u = f$ , then $- u$ is a weak subsolution of the same operator with coefficients $(a, b, c)$ and source $- f$ , by linearity of the form; applying step 1.1 or step 2.1 yields the stated lower-bound versions with $u^{-} = (- u)^{+}$ and $f^{-}$ . If $u$ is a weak solution of $L u = 0$ with $b = c = 0$ , let $s : = \sup_{\partial Ω} u$ . For every finite a.e. upper bound $t$ on $u$ , $(u - t)^{+} = 0$ , so [F1] implies $T u \leq t$ a.e.; taking infima gives $s \leq {ess sup}_{Ω} u$ . If $s = + \infty$ , this forces ${ess sup}_{Ω} u = + \infty = s$ . If $s$ is finite, $(u - s)^{+} \in H_{0}^{1} (Ω)$ , and density extends the weak identity to this test. Since $b = c = 0$ , it gives $0 = a (u, (u - s)^{+}) = \int a^{i j} D_{j} (u - s)^{+} D_{i} (u - s)^{+}$ , so $u \leq s$ a.e. The reverse trace bound just proved gives $s \leq {ess sup}_{Ω} u$ , and hence ${ess sup}_{Ω} u = s$ .

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Weak comparison and uniqueness for the Dirichlet problem

Statement

Assume Countable Choice and the Axiom of Choice. Let $n \geq 2$ , let $Ω \subset R^{n}$ be a bounded $C^{1}$ domain, and let $L, a$ be as in Weak subsolutions and supersolutions of a divergence-form equation with real $L^{\infty}$ coefficients, ellipticity $θ$ and bounds $M_{a}, M_{b}, M_{c}$ , satisfying $c \geq 0$ a.e. and the weak sign condition of Weak maximum principle for coercive divergence-form equations. Let $f \in L_{l o c}^{1} (Ω)$ and let $u, v \in H^{1} (Ω; R)$ be a local weak subsolution resp. weak supersolution of $L u = f$ with $u \leq v$ on $\partial Ω$ , i.e. $(u - v)^{+} \in H_{0}^{1} (Ω)$ . Then $u \leq v$ a.e. on $Ω$ . Consequently:

if $b \equiv 0$ , $c \geq 0$ a.e. and $f \in L^{q} (Ω)$ with $q > n / 2$ , then every weak solution of $L u = f$ with $u \leq 0$ on $\partial Ω$ satisfies ${ess sup}_{Ω} u \leq C ∥ f^{+} ∥_{L^{q} (Ω)}$ with the constant of Weak maximum principle for coercive divergence-form equations;
if $b \equiv 0$ , $c \geq 0$ a.e. and $f = 0$ , then two weak solutions of $L u = 0$ with the same trace in $H^{1 / 2} (\partial Ω)$ (Weak Dirichlet solutions for a divergence-form operator) agree a.e. on $Ω$ ; in particular the homogeneous Dirichlet problem has at most one weak solution for each admissible boundary datum.

Facts & Assumptions

Given: Countable Choice and the Axiom of Choice; a bounded $C^{1}$ domain $Ω \subset R^{n}$ , $n \geq 2$ ; real coefficients satisfying $c \geq 0$ and the weak-sign hypotheses of Weak maximum principle for coercive divergence-form equations; a datum $f \in L_{l o c}^{1} (Ω)$ ; and a local weak subsolution $u$ and local weak supersolution $v$ of $L u = f$ with $(u - v)^{+} \in H_{0}^{1} (Ω)$ .

[F1]

Linearity of the form: for every real nonnegative $φ \in C_{c}^{\infty} (Ω)$ , $a (u - v, φ) = a (u, φ) - a (v, φ)$ ; the form is the one of Uniformly elliptic divergence-form operators and their sesquilinear forms.

[F2]

Weak maximum principle: under $c \geq 0$ and the weak-sign condition, a real local weak subsolution $W \in H^{1} (Ω)$ of $L W = 0$ satisfies ${ess sup}_{Ω} W \leq \sup_{\partial Ω} W^{+}$ ; with $b = 0, c \geq 0$ and $g \in L^{q} (Ω)$ , $q > n / 2$ , a local subsolution with $W^{+} \in H_{0}^{1}$ satisfies ${ess sup}_{Ω} W \leq C ∥ g^{+} ∥_{L^{q}}$ (Weak maximum principle for coercive divergence-form equations).

[F3]

Boundary order and traces: $\sup_{\partial Ω} W = {ess sup}_{\partial Ω} T W$ , and $W^{+} \in H_{0}^{1} (Ω)$ implies $\sup_{\partial Ω} W^{+} = 0$ ; moreover $(u - v)^{+} \in H_{0}^{1} (Ω)$ is exactly the boundary inequality $u \leq v$ on $\partial Ω$ (A function whose trace is at most a level has positive part in the zero-boundary space, Weak subsolutions and supersolutions of a divergence-form equation, The kernel of the trace is the closure of the test functions).

Proof

technique · direct; apply the linearity of the form to the difference and invoke the weak maximum principle

1.1givenF1F3

The difference is a local weak subsolution of the homogeneous equation. Let $W : = u - v \in H^{1} (Ω; R)$ and let $φ \in C_{c}^{\infty} (Ω; R)$ be nonnegative. The local subsolution and supersolution inequalities give $a (u, φ) \leq \int_{Ω} f φ$ and $a (v, φ) \geq \int_{Ω} f φ$ , hence by [F1] $a (W, φ) \leq 0$ . Moreover $(u - v)^{+} = W^{+} \in H_{0}^{1} (Ω)$ by hypothesis, so $\sup_{\partial Ω} W^{+} = 0$ by [F3].

2.1step 1.1F2F3

Conclusion of the comparison. Step 1.1 exhibits $W$ as a weak subsolution of $L W = 0$ whose positive part lies in $H_{0}^{1} (Ω)$ ; [F2] gives ${ess sup}_{Ω} W \leq \sup_{\partial Ω} W^{+} = 0$ , that is, $u \leq v$ a.e. on $Ω$ .

3.1step 2.1F2F3

Consequence 1. If $b \equiv 0$ , $c \geq 0$ and $f \in L^{q} (Ω)$ with $q > n / 2$ , and $u$ is a weak solution with $u \leq 0$ on $\partial Ω$ , then $u$ is a weak subsolution of $L u = f$ and $u^{+} \in H_{0}^{1} (Ω)$ by the boundary hypothesis; the forcing clause of [F2] gives ${ess sup}_{Ω} u \leq C ∥ f^{+} ∥_{L^{q} (Ω)}$ with the constant recorded in Weak maximum principle for coercive divergence-form equations.

4.1step 2.1F3∎

Consequence 2 (uniqueness). Let $u, v$ be weak solutions of $L u = 0$ with the same trace in $H^{1 / 2} (\partial Ω)$ . Then $(u - v)^{+} \in H_{0}^{1} (Ω)$ and $(v - u)^{+} \in H_{0}^{1} (Ω)$ because the traces agree (The kernel of the trace is the closure of the test functions), so step 2.1 applied to the pair $(u, v)$ and to $(v, u)$ gives $u \leq v$ and $v \leq u$ a.e., i.e. $u = v$ a.e. Hence the homogeneous Dirichlet problem has at most one weak solution for each admissible boundary datum, and the comparison statement and its two consequences use only the declared choice principles.

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De Giorgi local boundedness of homogeneous subsolutions

Statement

Assume Countable Choice and the Axiom of Choice. Let $n \geq 2$ , let $Ω \subseteq R^{n}$ be open, let $0 < θ \leq M_{a}^{2}$ , let $A = (a^{i j})$ be measurable symmetric with $θ ∣ ξ ∣^{2} \leq \sum a^{i j} (x) ξ_{i} ξ_{j} \leq M_{a}^{2} ∣ ξ ∣^{2}$ for a.e. $x$ and all $ξ$ , and let $L_{0} u = - D_{i} (a^{i j} D_{j} u)$ . Let $u \in H^{1} (Ω; R)$ satisfy $u \geq 0$ a.e. and $a_{0} (u, v) \leq 0 for every v \in H_{0}^{1} (Ω), v \geq 0 a . e .,$ i.e. $u$ is a nonnegative weak subsolution of $L_{0} u = 0$ (Weak subsolutions and supersolutions of a divergence-form equation). Then $u$ is locally bounded, and for every ball $B_{R} (x_{0}) ⋐ Ω$ , every $0 < ρ < 1$ and every $p > 0$ , ${ess sup}_{B_{ρ R} (x_{0})} u \leq C (\frac{1}{∣ B_{R} (x_{0}) ∣} \int_{B_{R} (x_{0})} u^{p} d x)^{1 / p}, C = C (n, θ, M_{a}, ρ, p) .$ For $n = 2$ the same statement holds with the critical Sobolev embedding in place of the $2^{*}$ embedding. The constant is scale invariant: it does not depend on $R$ or $x_{0}$ .

Facts & Assumptions

Given: Countable Choice and the Axiom of Choice; an open $Ω \subseteq R^{n}$ , $n \geq 2$ ; constants $0 < θ \leq M_{a}^{2}$ ; a measurable symmetric coefficient field $A$ with $θ ∣ ξ ∣^{2} \leq ⟨ A ξ, ξ ⟩ \leq M_{a}^{2} ∣ ξ ∣^{2}$ a.e.; a nonnegative class $u \in H^{1} (Ω; R)$ with $a_{0} (u, v) \leq 0$ for every nonnegative $v \in H_{0}^{1} (Ω)$ ; a ball $B_{R} (x_{0}) ⋐ Ω$ .

[F1]

Assume Countable Choice and the Axiom of Choice. $a_{0} (w, v) = \int_{Ω} a^{i j} D_{j} w D_{i} v d x$ is defined for $w, v \in H^{1} (Ω; R)$ , and the subsolution inequality is the one of Weak subsolutions and supersolutions of a divergence-form equation with $f = 0$ ; for $η \in C_{c}^{\infty} (Ω)$ and $k \in R$ the class $η^{2} (u - k)^{+}$ is an admissible nonnegative test (Positive-part truncation calculus and admissible cut-off weak tests).

[F2]

Assume Countable Choice and the Axiom of Choice. Truncated Caccioppoli estimate: for $k \in R$ , $f = 0$ and concentric balls $B_{r} ⋐ B_{R}$ , $\int_{B_{r}} ∣ D (u - k)^{+} ∣^{2} \leq C_{0} (R - r)^{- 2} \int_{B_{R}} (u - k)^{+ 2}$ with $C_{0} = C_{0} (θ, M_{a})$ (Caccioppoli inequality for truncated subsolutions).

[F3]

Assume Countable Choice and the Axiom of Choice. Level-set step: if $u \in H^{1} (B_{R})$ and $\int_{B_{ρ}} ∣ D (u - k)^{+} ∣^{2} \leq C_{0} (R - ρ)^{- 2} \int_{B_{R}} (u - k)^{+ 2}$ holds for all $0 < ρ < R$ and all levels $k$ , then for $n \geq 3$ , $\int_{B_{r}} (u - k)^{+ 2} \leq C (n, C_{0}) (R - r)^{- 2} (k - h)^{- 4 / n} (\int_{B_{R}} (u - h)^{+ 2})^{1 + 2 / n}$ . For $n = 2$ and each $0 < δ < 1$ , the power and integral exponent use $2 δ$ and $1 + δ$ , and the radius factor is $R^{2 - 2 δ} (R - r)^{- 2}$ ; the constant may depend on $δ$ (Sobolev level-set step: energy decay with explicit level gap and radius loss).

[F4]

Nonlinear iteration: if $δ > 0$ , $C \geq 1$ , $B \geq 1$ and $Y_{j + 1} \leq C B^{j} Y_{j}^{1 + δ}$ with $Y_{0} \leq C^{- 1 / δ} (2 B)^{- 1 / δ^{2}}$ , then $Y_{j} \leq Y_{0} λ^{j} \to 0$ with $λ = (2 B)^{- 1 / δ}$ (The nonlinear geometric iteration: an explicit threshold forces convergence to zero).

[F5]

Essential supremum and $L^{p}$ means: a class $w$ satisfies $w \leq T$ a.e. if and only if $ess sup w \leq T$ . For every $0 < p < q$ , Hölder applied to $∣ w ∣^{p}$ and $1$ with exponents $q / p$ and $q / (q - p)$ gives $\int_{E} ∣ w ∣^{p} \leq ∣ E ∣^{1 - p / q} (\int_{E} ∣ w ∣^{q})^{p / q}$ (The essential supremum is attained as the least essential bound, The essential supremum of a measurable function with respect to a measure, Holder's inequality for integrals, including the endpoint cases, The space $L^{p} (μ)$ as the quotient by null functions, The average of a locally integrable function over a Euclidean ball).

[F6]

Assume the Axiom of Choice. The globally Lipschitz chain rule and weak product rule justify the compositions and cutoff tests. For a convex Lipschitz truncation $P_{N}$ , scalar convolution followed by subtracting the value at zero gives smooth convex nondecreasing approximants; their compositions converge in $H_{l o c}^{1}$ by the chain rule and dominated convergence. Monotone convergence applies to $P_{N} (u) ↑ u^{β}$ as $N \to \infty$ (Chain rule for globally Lipschitz scalar maps of Sobolev functions, Weak Leibniz rule with a smooth factor, Dominated convergence, Monotone convergence for the integral).

[F7]

Weighted Young inequality: if $0 < p < 2$ , then for $X, Y \geq 0$ and every $ϵ > 0$ , $X Y \leq ϵ X^{2 / (2 - p)} + C_{p} ϵ^{- (2 - p) / p} Y^{2 / p}$ , with $C_{p}$ depending only on $p$ (Young's inequality for conjugate real exponents).

Proof

technique · derive the $L^2$ mean-to-supremum estimate by the dyadic De Giorgi level recurrence, obtain any smaller-ball ratio by a finite cover with an explicit radius-gap constant, then use convex power truncations for $p\ge2$ and a two-scale interpolation iteration for $0<p<2$

1.1givenF1F6algebra

Convex power truncations. Fix $β \geq 1$ and $N \geq 1$ , and define the convex nondecreasing Lipschitz function $P_{N} (s) : = {\begin{cases} 0, & s \leq 0, \\ s^{β}, & 0 < s \leq N, \\ N^{β} + β N^{β - 1} (s - N), & s > N . \end{cases}$ It satisfies $P_{N} (0) = 0$ and $P_{N} (u) \geq 0$ since $u \geq 0$ . Let $G_{ϵ}$ be a smooth convolution of $P_{N}$ minus its value at zero. Then $G_{ϵ} (0) = 0$ , $G_{ϵ}^{'} \geq 0$ , $G_{ϵ}^{''} \geq 0$ , and the Lipschitz constants are uniformly bounded for this fixed $N$ . For a nonnegative $ϕ \in C_{c}^{\infty} (Ω)$ , the test $G_{ϵ}^{'} (u) ϕ$ is nonnegative and belongs to $H_{0}^{1}$ on a bounded neighborhood of its support. Since the equation is homogeneous, density extends the subsolution inequality to this test. The chain and product rules give $a_{0} (G_{ϵ} (u), ϕ) = a_{0} (u, G_{ϵ}^{'} (u) ϕ) - \int_{Ω} G_{ϵ}^{''} (u) a^{i j} D_{j} u D_{i} u ϕ d x \leq 0.$ As $ϵ ↓ 0$ , the compositions converge to $P_{N} (u)$ in $H_{l o c}^{1}$ by [F6], so the displayed inequality passes to $P_{N} (u)$ against each smooth nonnegative test. Thus $P_{N} (u)$ is a nonnegative local weak subsolution. No subsolution property of the smooth approximants is required.

1.2givenF2F3algebra

The dyadic recurrence. Assume $\int_{B_{R}} u^{2} > 0$ (otherwise $u = 0$ a.e. on $B_{R}$ ), fix $B_{R} = B_{R} (x_{0}) ⋐ Ω$ and $T > 0$ , and put $k_{j} : = T (1 - 2^{- j})$ , $r_{j} : = R (1 / 2 + 2^{- j - 1})$ , $Y_{j} : = \int_{B_{r_{j}}} (u - k_{j})^{+ 2} d x$ for $j \geq 0$ . Set $δ : = 2 / n$ if $n \geq 3$ , and $δ : = 1 / 2$ if $n = 2$ (so the latter uses the finite exponent $κ = 4$ ). Applying [F3] with outer radius $r_{j}$ and inner radius $r_{j + 1}$ , and using $r_{j} - r_{j + 1} = R 2^{- j - 2}$ and $k_{j + 1} - k_{j} = T 2^{- j - 1}$ , gives $Y_{j + 1} \leq C_{1} B_{0}^{j} R^{- n δ} T^{- 2 δ} Y_{j}^{1 + δ}, B_{0} : = 2^{2 + 2 δ},$ where $C_{1} = C_{1} (n, θ, M_{a}) \geq 1$ . For $n = 2$ , the scaled radius factor in [F3] contributes $r_{j}^{2 - 2 δ} (r_{j} - r_{j + 1})^{- 2} \leq C R^{- 2 δ} 2^{2 j}$ ; for $n \geq 3$ , $n δ = 2$ and the same displayed scale follows directly.

2.1step 1.2F4F5

The iteration closes. Write $Z_{j} : = R^{- n} T^{- 2} Y_{j}$ . Then the recurrence of step 1.2 reads $Z_{j + 1} \leq C_{1} B_{0}^{j} Z_{j}^{1 + δ}$ , with $δ = 2 / n$ for $n \geq 3$ and $δ = 1 / 2$ for $n = 2$ , and $Z_{0} = R^{- n} T^{- 2} \int_{B_{R}} u^{2}$ . By [F4], if $R^{- n} T^{- 2} \int_{B_{R}} u^{2} \leq C_{1}^{- 1 / δ} (2 B_{0})^{- 1 / δ^{2}}$ then $Z_{j} \to 0$ ; choosing $T : = c_{0} (R^{- n} \int_{B_{R}} u^{2})^{1 / 2}$ with $c_{0} : = C_{1}^{1 / (2 δ)} (2 B_{0})^{1 / (2 δ^{2})}$ meets this condition. Then $\int_{B_{R / 2}} (u - T)^{+ 2} \leq Y_{j} \to 0$ , so $u \leq T$ a.e. on $B_{R / 2} (x_{0})$ and hence, by [F5], ${ess sup}_{B_{R / 2} (x_{0})} u \leq c_{0} (R^{- n} \int_{B_{R} (x_{0})} u^{2})^{1 / 2} = C_{2} (\frac{1}{∣ B_{R} (x_{0}) ∣} \int_{B_{R} (x_{0})} u^{2})^{1 / 2}$ with $C_{2} = C_{2} (n, θ, M_{a})$ .

3.1step 2.1algebra

Every smaller-ball ratio with a gap bound. Fix $0 < σ < 1$ and set $d : = (1 - σ) R / 2$ . A finite collection of balls $B_{d / 2} (x_{ℓ})$ with centers in $B_{σ R} (x_{0})$ covers $B_{σ R} (x_{0})$ , and each outer ball $B_{d} (x_{ℓ})$ is compactly contained in $B_{R} (x_{0})$ . Applying the half-ball $L^{2}$ estimate of step 2.1 to each outer ball yields ${ess sup}_{B_{d / 2} (x_{ℓ})} u \leq C_{2} {(\frac{1}{∣ B_{d} ∣} \int_{B_{d} (x_{ℓ})} u^{2})}^{1 / 2} \leq C_{2} {(\frac{2}{1 - σ})}^{n / 2} {(\frac{1}{∣ B_{R} ∣} \int_{B_{R}} u^{2})}^{1 / 2} .$ Taking the finite union gives the same bound on $B_{σ R}$ . This quantitative gap dependence controls the radius losses in the subsequent small-exponent argument. In particular, $u$ is essentially bounded on each strictly smaller ball.

4.1step 1.1step 3.1F6

The case $p \geq 2$ . If $\int_{B_{R}} u^{p} = \infty$ the estimate is automatic. Fix $p \geq 2$ and put $β : = p / 2$ . For each $N \geq 1$ , $w_{N} : = P_{N} (u)$ is a nonnegative local weak subsolution by step 1.1 and lies in $H^{1} (Ω)$ because $P_{N}$ is globally Lipschitz with $P_{N} (0) = 0$ . The zero-source inequality extends to all nonnegative $H_{0}^{1} (Ω)$ tests, so the local boundedness theorem applies. The arbitrary-ratio $p = 2$ estimate of step 3.1 gives ${ess sup}_{B_{ρ R}} w_{N} \leq C_{2} (ρ) (\frac{1}{∣ B_{R} ∣} \int_{B_{R}} w_{N}^{2})^{1 / 2}$ . As $N \to \infty$ , $w_{N} ↑ u^{β}$ and $w_{N}^{2} ↑ u^{2 β}$ , so monotone convergence [F6] and monotonicity of essential supremum give ${ess sup}_{B_{ρ R}} u^{β} \leq C_{2} (ρ) (\frac{1}{∣ B_{R} ∣} \int_{B_{R}} u^{p})^{1 / 2}$ . Taking the $β$ -th root proves the estimate, with constant $C_{2} (ρ)^{1 / β}$ .

4.2step 3.1F5F7algebra

The case $0 < p < 2$ . Put $A : = (\frac{1}{∣ B_{R} ∣} \int_{B_{R}} u^{p})^{1 / p}$ . If $A = 0$ , then $u = 0$ a.e. on $B_{R}$ ; otherwise $0 < A < \infty$ . Let $s_{*} : = (1 + ρ) / 2$ , $r_{j} : = ρ R + (s_{*} R - ρ R) (1 - 2^{- j})$ , and $M_{j} : = {ess sup}_{B_{r_{j}}} u$ . By step 3.1, $M_{j} \leq C_{*} (1 - r_{j} / R)^{- n / 2} (\frac{1}{∣ B_{R} ∣} \int_{B_{R}} u^{2})^{1 / 2} < \infty$ . Apply the $p = 2$ estimate of step 3.1 to $u$ on the outer ball $B_{r_{j + 1}}$ with inner ratio $r_{j} / r_{j + 1}$ . Its explicit gap bound gives a constant $C_{j} \leq C_{4} b^{j}$ (because $r_{j + 1} - r_{j}$ is a fixed multiple of $2^{- j} R$ ), and Holder gives $M_{j} \leq C_{j} M_{j + 1}^{1 - p / 2} A^{p / 2} .$ For any $ϵ > 0$ , [F7] yields $M_{j} \leq ϵ M_{j + 1} + C_{5} ϵ^{- (2 - p) / p} C_{j}^{2 / p} A$ . Choose $ϵ$ with $ϵ b^{2 / p} < 1$ and iterate. The geometric series $\sum_{j \geq 0} ϵ^{j} C_{j}^{2 / p}$ converges, while $M_{j} \leq M_{*} < \infty$ by step 3.1 on the fixed ball $B_{s_{*} R}$ , so $ϵ^{j} M_{j} \to 0$ . Hence $M_{0} \leq C_{6} A$ , proving the desired estimate on $B_{ρ R}$ . This proves every $0 < p < 2$ directly and requires no limit as the radius approaches $R$ .

5.1step 4.1step 4.2∎

Conclusion. Steps 4.1 and 4.2 prove the estimate for every $p > 0$ ; the constants depend only on $n, θ, M_{a}, ρ, p$ , and scaling shows independence of $R$ and $x_{0}$ .

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De Giorgi local boundedness with a scale-correct forcing term

Statement

Assume Countable Choice and the Axiom of Choice. Let $n \geq 2$ , let $Ω \subseteq R^{n}$ be open, let $A$ and $L_{0}$ be as in De Giorgi local boundedness of homogeneous subsolutions, and let $f \in L_{l o c}^{q} (Ω)$ with $q > n / 2$ . Let $u \in H^{1} (Ω; R)$ satisfy $u \geq 0$ a.e. and $a_{0} (u, φ) \leq \int_{Ω} f φ d x for every nonnegative φ \in C_{c}^{\infty} (Ω; R) .$ Then for every ball $B_{R} (x_{0}) ⋐ Ω$ , every $0 < ρ < 1$ and every $p > 0$ , ${ess sup}_{B_{ρ R} (x_{0})} u \leq C [(\frac{1}{∣ B_{R} (x_{0}) ∣} \int_{B_{R} (x_{0})} u^{p} d x)^{1 / p} + R^{2 - n / q} ∥ f^{+} ∥_{L^{q} (B_{R} (x_{0}))}],$ with $C = C (n, q, θ, M_{a}, ρ, p)$ independent of $R$ and $x_{0}$ . The factor $R^{2 - n / q}$ is dictated by dilation of the equation: the forcing term has the dimension of $u$ for every $q$ . The strict threshold $q > n / 2$ is an integrability hypothesis for this boundedness estimate, not a condition for dimensional consistency; the $n = 2$ case admits any $q > 1$ with the critical Sobolev embedding in place of the $2^{*}$ embedding.

Facts & Assumptions

Given: Countable Choice and the Axiom of Choice; an open set $Ω \subseteq R^{n}$ , $n \geq 2$ ; constants $0 < θ \leq M_{a}^{2}$ ; measurable symmetric coefficients $A$ with $θ ∣ ξ ∣^{2} \leq ⟨ A (x) ξ, ξ ⟩ \leq M_{a}^{2} ∣ ξ ∣^{2}$ ; the principal operator $L_{0} u = - D_{i} (a^{i j} D_{j} u)$ with form $a_{0}$ ; a source $f \in L_{l o c}^{q} (Ω)$ , $q > n / 2$ ; a nonnegative $u \in H^{1} (Ω; R)$ with $a_{0} (u, φ) \leq \int_{Ω} f φ d x$ for all nonnegative $φ \in C_{c}^{\infty} (Ω; R)$ ; a ball $B_{R} (x_{0}) ⋐ Ω$ and $0 < ρ < 1$ .

[F1]

Assume Countable Choice and the Axiom of Choice. Sobolev positive-part calculus: for $w \in H^{1} (Ω; R)$ , $w^{+} \in H^{1}$ with $D w^{+} = 1_{{w > 0}} D w$ and $D w = 0$ a.e. on ${w = 0}$ ; multiplication by a compactly supported smooth factor obeys the weak product rule. If $a_{0} (w, v) \leq 0$ for every nonnegative $v \in H_{0}^{1} (Ω)$ , then $w^{+}$ is a weak subsolution of the principal operator, so $a_{0} (w^{+}, v) \leq 0$ for every nonnegative $v \in H_{0}^{1} (Ω)$ . Here is the admissible truncation proof. For $ϕ \in C_{c}^{\infty} (Ω)$ , $ϕ \geq 0$ , set $η_{ϵ} (t) = \min {1, t^{+} / ϵ}$ . The chain and product rules give $ψ_{ϵ} = ϕ η_{ϵ} (w) \in H^{1}$ with compact support in $Ω$ , hence $ψ_{ϵ} \in H_{0}^{1} (Ω)$ ; it is nonnegative, and $D_{i} ψ_{ϵ} = η_{ϵ} (w) D_{i} ϕ + ϵ^{- 1} ϕ 1_{{0 < w < ϵ}} D_{i} w$ (the level-set endpoints contribute zero because Sobolev gradients vanish a.e. on a level set). Thus $a_{0} (w, ψ_{ϵ}) = \int_{Ω} η_{ϵ} (w) a^{i j} D_{j} w D_{i} ϕ d x + \frac{1}{ϵ} \int_{{0 < w < ϵ}} ϕ a^{i j} D_{j} w D_{i} w d x \leq 0.$ The second integral is nonnegative by symmetry, ellipticity, and $ϕ \geq 0$ , so the first is nonpositive. As $ϵ ↓ 0$ , $η_{ϵ} (w) \to 1_{{w > 0}}$ pointwise and is bounded by $1$ ; dominated convergence applies because $A$ is bounded and $∣ D w ∣ ∣ D ϕ ∣$ is integrable on $supp ϕ$ . Using $D w^{+} = 1_{{w > 0}} D w$ gives $a_{0} (w^{+}, ϕ) \leq 0$ . To extend from smooth tests to every nonnegative $v \in H_{0}^{1}$ , choose $z_{j} \in C_{c}^{\infty} (Ω)$ with $z_{j} \to v$ in $H^{1}$ and pass to a subsequence with $z_{j} \to v$ a.e. The positive-part gradient formulas give $D z_{j}^{+} - D v = 1_{{z_{j} > 0}} (D z_{j} - D v) + (1_{{z_{j} > 0}} - 1_{{v > 0}}) D v .$ The first term tends to zero in $L^{2}$ ; the second does too by dominated convergence, since its indicator tends to zero on ${v > 0}$ and $D v = 0$ a.e. on ${v = 0}$ . Also $z_{j}^{+} \to v$ in $L^{2}$ by the $1$ -Lipschitz property, hence $z_{j}^{+} \to v$ in $H^{1}$ . Each nonzero $z_{j}^{+}$ has compact support in $Ω$ ; zero-extend it and convolve with a nonnegative unit-mass radial mollifier, chosen with support radius smaller than $dist (supp z_{j}^{+}, \partial Ω)$ (if $z_{j}^{+} = 0$ , keep the zero function). The mollified functions are nonnegative and in $C_{c}^{\infty} (Ω)$ , and converge to $z_{j}^{+}$ in $H^{1}$ by approximate-identity convergence applied to the function and its weak gradient. A diagonal choice gives nonnegative smooth tests converging to $v$ . Boundedness of $a_{0} (w, \cdot)$ passes the inequality to $v$ . In particular, no product of an indicator with an arbitrary test is asserted to lie in $H_{0}^{1}$ . (Positive-part truncation calculus and admissible cut-off weak tests, Positive, negative, and truncated Sobolev functions, Chain rule for globally Lipschitz scalar maps of Sobolev functions, Weak Leibniz rule with a smooth factor, Compactly supported Sobolev functions extend by zero in every integer order, A radial mollifier family in Rn, A smooth bump between concentric Euclidean balls, The mollifier family generated by a unit-mass smooth bump, Interior mollification commutes with weak derivatives, Every $L^{1}$ approximate identity converges to the identity in $L^{p}$ for $1 \leq p < \infty$ , Dominated convergence, Zero-boundary Sobolev space as a norm closure, Integer-order Sobolev spaces and their norms, The elliptic form is well defined and bounded on $H^{1}$ , Uniformly elliptic divergence-form operators and their sesquilinear forms, Weak subsolutions and supersolutions of a divergence-form equation).

[F2]

Assume the Axiom of Choice. Lax-Milgram and coercivity: $H^{1} (B_{1})$ is Hilbert by $H^{k}$ is a Hilbert space under the derivative-sum inner product. Its subspace $H_{0}^{1} (B_{1})$ is closed and linear by its closure definition: a Cauchy sequence there converges in $H^{1} (B_{1})$ , and its limit still lies in the closure of the smooth tests. Thus the restricted derivative-sum inner product makes $H_{0}^{1} (B_{1})$ Hilbert. The form $a_{0}$ is a bounded sesquilinear form on the Hilbert space $H_{0}^{1} (B_{1})$ with coercivity constant $α = θ / (1 + C_{P} (B_{1})^{2})$ , and for every bounded conjugate-linear functional $G$ on $H_{0}^{1} (B_{1})$ there is a unique $h \in H_{0}^{1} (B_{1})$ with $a_{0} (h, v) = G (v)$ for all $v \in H_{0}^{1} (B_{1})$ , the weak Dirichlet solution; it satisfies $α ∥ h ∥_{H_{0}^{1}} \leq ∥ G ∥_{H^{- 1}}$ (The Lax--Milgram theorem, Coercivity of the principal Dirichlet form, The elliptic form is well defined and bounded on $H^{1}$ , The negative Sobolev space $H^{- 1} (Ω)$ , Weak Dirichlet solutions for a divergence-form operator, Zero-boundary Sobolev space as a norm closure).

[F3]

Assume the Axiom of Choice. Embedding of $H_{0}^{1}$ into the dual exponents of $L^{q}$ : for $n \geq 3$ and $1 \leq r < 2^{*} = 2 n / (n - 2)$ there is $C_{r}$ with $∥ w ∥_{L^{r} (B_{1})} \leq C_{r} ∥ w ∥_{H_{0}^{1} (B_{1})}$ for all $w \in H_{0}^{1} (B_{1})$ , by Holder and the Sobolev inequality; for $n = 2$ the same holds for every finite $r$ by the critical embedding $W_{0}^{1, 2} (B_{1}) ↪ L^{r} (B_{1})$ (The Sobolev inequality for zero-boundary Sobolev closures on open sets, The critical Sobolev embedding into every finite $L^{q}$ , Holder's inequality for integrals, including the endpoint cases, The space $L^{p} (μ)$ as the quotient by null functions).

[F4]

Assume the Axiom of Choice. The weak maximum principle with a source on the bounded $C^{1}$ domain $B_{1}$ : if $v \in H^{1} (B_{1}; R)$ satisfies $a_{0} (v, φ) \leq \int_{B_{1}} g φ d x$ for every nonnegative $φ \in H_{0}^{1} (B_{1})$ with $g \in L^{q} (B_{1})$ , $q > n / 2$ , then ${ess sup}_{B_{1}} v \leq \sup_{\partial B_{1}} v^{+} + C_{0} ∥ g^{+} ∥_{L^{q} (B_{1})}$ with $C_{0} = C_{0} (n, q, θ, M_{a}, B_{1})$ ; in particular for $v \in H_{0}^{1} (B_{1})$ one has ${ess sup}_{B_{1}} v \leq C_{0} ∥ g^{+} ∥_{L^{q} (B_{1})}$ , and if $g \leq 0$ then $v \leq 0$ a.e. (Weak maximum principle for coercive divergence-form equations, The $L^{p}$ trace operator on a bounded $C^{1}$ domain, The kernel of the trace is the closure of the test functions, Bounded C^k domains and boundary charts).

[F5]

Assume the Axiom of Choice. Homogeneous local boundedness: if $w \in H^{1} (B_{1}; R)$ satisfies $w \geq 0$ a.e. and $a_{0} (w, v) \leq 0$ for every nonnegative $v \in H_{0}^{1} (B_{1})$ , then for every $0 < s < 1$ , every $0 < σ < 1$ and every $p > 0$ , ${ess sup}_{B_{σ s}} w \leq C_{1} (\frac{1}{∣ B_{s} ∣} \int_{B_{s}} w^{p} d x)^{1 / p}$ ; this is the homogeneous theorem applied on the compactly contained ball $B_{s} ⋐ B_{1}$ (De Giorgi local boundedness of homogeneous subsolutions).

[F6]

Assume the Axiom of Choice. Poincare inequality on $W_{0}^{1, 2}$ (The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction, The essential supremum of a measurable function with respect to a measure).

Proof

technique · direct; rescale the problem to the unit ball, remove the source by subtracting a barrier built with Lax-Milgram whose size is controlled by the weak maximum principle, apply the homogeneous local boundedness estimate to the positive part of the difference, and undo the rescalings

1.1givenalgebra

Scaling the problem to the unit ball. Define $v (y) : = u (x_{0} + R y)$ for $y \in B_{1}$ , $A_{R} (y) : = A (x_{0} + R y)$ and $g (y) : = R^{2} f (x_{0} + R y)$ , so that $g \in L^{q} (B_{1})$ with $∥ g^{+} ∥_{L^{q} (B_{1})} = R^{2 - n / q} ∥ f^{+} ∥_{L^{q} (B_{R} (x_{0}))}$ and $\frac{1}{∣ B_{1} ∣} \int_{B_{1}} v^{p} d y = \frac{1}{∣ B_{R} (x_{0}) ∣} \int_{B_{R} (x_{0})} u^{p} d x$ by the change of variables $x = x_{0} + R y$ ; the coefficients $A_{R}$ are again measurable, symmetric and uniformly elliptic with the same constants $θ, M_{a}$ . Since $f \in L^{q} (B_{R})$ and $q > n / 2$ , Sobolev and Holder extend the local inequality by density to nonnegative $H_{0}^{1} (B_{R})$ tests. For every such test $φ \in H_{0}^{1} (B_{1})$ the pullback $φ_{R} (x) : = φ ((x - x_{0}) / R)$ lies in $H_{0}^{1} (B_{R} (x_{0}))$ with $D φ_{R} (x) = R^{- 1} D φ (y)$ , hence $a_{0}^{R} (v, φ) = R^{2 - n} \int_{B_{R} (x_{0})} a^{i j} D_{j} u D_{i} φ_{R} d x \leq R^{2 - n} \int_{B_{R} (x_{0})} f φ_{R} d x = \int_{B_{1}} g φ d y$ , where $a_{0}^{R}$ is the form of $A_{R}$ ; so $v \geq 0$ satisfies the same subsolution inequality on $B_{1}$ with source $g$ .

2.1step 1.1F1F2F3F4F5

Removing a small source by a barrier. Let $U \in H^{1} (B_{1}; R)$ satisfy $U \geq 0$ a.e. and $a_{0} (U, φ) \leq \int_{B_{1}} g φ d x$ for all nonnegative $φ \in H_{0}^{1} (B_{1})$ with $∥ g^{+} ∥_{L^{q} (B_{1})} \leq 1$ ; then for every $0 < ρ < 1$ , every $p > 0$ and some $C_{2} = C_{2} (n, q, θ, M_{a}, ρ, p)$ one has ${ess sup}_{B_{ρ}} U \leq C_{2} (\frac{1}{∣ B_{1} ∣} \int_{B_{1}} U^{p} d x)^{1 / p} + C_{2}$ . Indeed, by [F3] the functional $φ \mapsto \int_{B_{1}} g^{+} φ d x$ is bounded on $H_{0}^{1} (B_{1})$ , so by [F2] there is a unique $h \in H_{0}^{1} (B_{1})$ with $a_{0} (h, φ) = \int_{B_{1}} g^{+} φ d x$ for all $φ \in H_{0}^{1} (B_{1})$ . Since $- h$ is a weak subsolution with source $- g^{+} \leq 0$ and zero boundary values, [F4] gives $h \geq 0$ ; it also gives ${ess sup}_{B_{1}} h \leq C_{0} ∥ g^{+} ∥_{L^{q} (B_{1})} \leq C_{0}$ . The difference $w : = U - h$ satisfies $a_{0} (w, φ) \leq - \int_{B_{1}} g^{-} φ d x \leq 0$ for all nonnegative $φ \in H_{0}^{1} (B_{1})$ , so $w^{+}$ is a nonnegative homogeneous weak subsolution by [F1]. Since $h \geq 0$ and $U \geq 0$ , $w^{+} = (U - h)^{+} \leq U$ . Fix $s : = (1 + ρ) / 2$ , so $ρ < s < 1$ , and apply [F5] to $w^{+}$ on the outer ball $B_{s} ⋐ B_{1}$ with inner ratio $σ = ρ / s$ . This gives ${ess sup}_{B_{ρ}} w^{+} \leq C (\frac{1}{∣ B_{s} ∣} \int_{B_{s}} (w^{+})^{p})^{1 / p} \leq C^{'} (\frac{1}{∣ B_{1} ∣} \int_{B_{1}} U^{p})^{1 / p}$ , where the volume ratio is absorbed into $C^{'}$ . Hence ${ess sup}_{B_{ρ}} U \leq {ess sup}_{B_{ρ}} w^{+} + {ess sup}_{B_{1}} h \leq C_{2} (\frac{1}{∣ B_{1} ∣} \int_{B_{1}} U^{p})^{1 / p} + C_{2}$ .

3.1step 1.1step 2.1F5F6algebra∎

Undoing the rescaling and the normalisation. If $∥ g^{+} ∥_{L^{q} (B_{1})} = 0$ then $v$ is itself a homogeneous weak subsolution and [F5] gives the claim directly with the forcing term absent. Otherwise put $F_{R} : = ∥ g^{+} ∥_{L^{q} (B_{1})} > 0$ and $U : = v / F_{R}$ , so that $U \geq 0$ , $a_{0} (U, φ) \leq \int_{B_{1}} (g / F_{R}) φ d x$ for all nonnegative $φ \in H_{0}^{1} (B_{1})$ and $∥ (g / F_{R})^{+} ∥_{L^{q} (B_{1})} = 1$ ; step 2.1 applied to $U$ gives ${ess sup}_{B_{ρ}} U \leq C_{2} (\frac{1}{∣ B_{1} ∣} \int_{B_{1}} U^{p})^{1 / p} + C_{2}$ , and multiplying by $F_{R}$ , ${ess sup}_{B_{ρ}} v \leq C_{2} (\frac{1}{∣ B_{1} ∣} \int_{B_{1}} v^{p})^{1 / p} + C_{2} F_{R}$ . Substituting the identities of step 1.1 gives ${ess sup}_{B_{ρ R} (x_{0})} u \leq C_{2} (\frac{1}{∣ B_{R} (x_{0}) ∣} \int_{B_{R} (x_{0})} u^{p} d x)^{1 / p} + C_{2} R^{2 - n / q} ∥ f^{+} ∥_{L^{q} (B_{R} (x_{0}))}$ , which is the asserted estimate with $C : = \max {C_{2}, 1}$ ; the constant depends only on $n, q, θ, M_{a}, ρ, p$ , and the argument uses Countable Choice and the Axiom of Choice exactly through the cited suppliers.

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De Giorgi oscillation reduction: one half-level set is small

Statement

Assume Countable Choice and the Axiom of Choice. Let $n \geq 2$ , let $Ω \subseteq R^{n}$ be open, let $A$ and $L_{0}$ be as in De Giorgi local boundedness of homogeneous subsolutions, and let $u \in H^{1} (Ω; R)$ be a weak solution of $L_{0} u = 0$ on $Ω$ . Write $M : = {ess sup}_{B_{R} (x_{0})} u$ , $m : = {ess inf}_{B_{R} (x_{0})} u$ and ${osc}_{B_{R} (x_{0})} u : = M - m$ for balls $B_{R} (x_{0}) ⋐ Ω$ . Then the two half-level sets cannot both be large, and one of them is small enough to reduce the oscillation:

(dichotomy) at least one of $∣ {u > (M + m) / 2} \cap B_{R} (x_{0}) ∣$ and $∣ {u < (M + m) / 2} \cap B_{R} (x_{0}) ∣$ is at most $\frac{1}{2} ∣ B_{R} (x_{0}) ∣$ ;
(quantitative reduction) there are constants $η = η (n, θ, M_{a}) \in (0, 1)$ and $C = C (n, θ, M_{a})$ such that for every $B_{R} (x_{0})$ with $B_{2 R} (x_{0}) ⋐ Ω$ , ${ess osc}_{B_{R / 2} (x_{0})} u \leq η {ess osc}_{B_{R} (x_{0})} u .$ Moreover the constant $η$ may be chosen as $1 - η_{0} / 2$ where $η_{0} > 0$ depends only on $n, θ, M_{a}$ ; the proof uses localized truncated Caccioppoli estimates and applies the local boundedness estimate De Giorgi local boundedness of homogeneous subsolutions to a nonnegative truncation on the inner ball.

Facts & Assumptions

Given: Countable Choice and the Axiom of Choice; an open $Ω \subseteq R^{n}$ , $n \geq 2$ ; a measurable symmetric coefficient field $A$ with $θ ∣ ξ ∣^{2} \leq ⟨ A ξ, ξ ⟩ \leq M_{a}^{2} ∣ ξ ∣^{2}$ a.e.; a weak solution $u \in H^{1} (Ω; R)$ of $L_{0} u = 0$ ; and a ball $B_{2 R} (x_{0}) ⋐ Ω$ .

[F1]

Assume Countable Choice and the Axiom of Choice. Local boundedness: every nonnegative weak subsolution $w$ of $L_{0} w = 0$ on an open set satisfies ${ess sup}_{B_{ρ r} (y)} w \leq C_{1} (ρ) (\frac{1}{∣ B_{r} (y) ∣} \int_{B_{r} (y)} w^{2})^{1 / 2}$ for every $B_{r} (y) ⋐ Ω$ and every $0 < ρ < 1$ , with $C_{1} (ρ) = C_{1} (n, θ, M_{a}, ρ)$ (De Giorgi local boundedness of homogeneous subsolutions).

[F2]

Truncated Caccioppoli estimate and truncation subsolution property. For a solution $u$ of $L_{0} u = 0$ and any $k$ , choose smooth nondecreasing $χ_{ϵ}$ with $χ_{ϵ} = 0$ on $(- \infty, 0]$ , $χ_{ϵ} = 1$ on $[ϵ, \infty)$ , and $χ_{ϵ} \geq 0$ . Testing the local equation with $φ χ_{ϵ} (u - k)$ for nonnegative $φ \in C_{c}^{\infty}$ is justified by $H_{0}^{1}$ density; expansion gives $0 = \int χ_{ϵ} (u - k) A D u \cdot D φ + \int φ χ_{ϵ}^{'} (u - k) A D u \cdot D u$ , so the first integral is nonpositive. Letting $ϵ ↓ 0$ , the Sobolev chain rule and $D u = 0$ a.e. on ${u = k}$ give $a_{0} ((u - k)^{+}, φ) \leq 0$ . Thus $(u - k)^{+}$ is a nonnegative local weak subsolution. Also, for $B_{r} ⋐ B_{R}$ , $\int_{B_{r}} ∣ D (u - k)^{+} ∣^{2} \leq C_{0} (R - r)^{- 2} \int_{B_{R}} (u - k)^{+ 2}$ with $C_{0} = C_{0} (θ, M_{a})$ (Local weak solutions of a divergence-form operator, Weak subsolutions and supersolutions of a divergence-form equation, Positive-part truncation calculus and admissible cut-off weak tests, Caccioppoli inequality for truncated subsolutions, Sobolev level-set step: energy decay with explicit level gap and radius loss).

[F3]

Assume the Axiom of Choice. Smooth functions on the closed ball are dense in $H^{1} (B_{R})$ , and $H^{1} (B_{R})$ is the closure of $C^{\infty} ({\overline{B}}_{R})$ under the Sobolev norm; a.e. convergence and $L^{2}$ convergence of the gradients may be assumed along a subsequence (Ambient smooth restrictions are dense on bounded C^k domains).

[F4]

Measure conventions: $ess sup$ and $ess inf$ are the least essential upper and greatest essential lower bounds, and $∣ \cdot ∣$ denotes Lebesgue measure. The signed-extrema convention is Weak subsolutions and supersolutions of a divergence-form equation; The essential supremum is attained as the least essential bound and The essential supremum of a measurable function with respect to a measure concern the corresponding absolute essential bound.

[F5]

Fatou's lemma: if nonnegative indicators have pointwise lower limit at least the indicator of a limiting set, then the measure of that set is at most the lower limit of the approximating measures (Fatou's lemma).

Proof

technique · direct; after normalisation the measure of the level sets is driven down by a telescoping De Giorgi iteration, and the local boundedness estimate turns the small measure of the top level set into a sup bound

1.1givenF1F2F4algebra

Dichotomy and normalisation. For any $B_{R} (x_{0}) ⋐ Ω$ , local boundedness applied to $u^{+}$ and $(- u)^{+}$ on slightly larger interior balls gives finite $M, m$ ; these truncations are subsolutions by [F2]. The strict sets ${u > (M + m) / 2} \cap B_{R}$ and ${u < (M + m) / 2} \cap B_{R}$ are disjoint, so at least one has measure at most $∣ B_{R} ∣ / 2$ , proving claim 1. For claim 2 assume now $B_{2 R} (x_{0}) ⋐ Ω$ . If $L : = (M - m) / 2 = 0$ , then $u$ is constant a.e. on $B_{R}$ and the reduction is immediate. Otherwise define $v (y) : = (u (x_{0} + R y) - (M + m) / 2) / L$ on $B_{2} (0)$ . It solves the homogeneous equation with rescaled coefficients $A (x_{0} + R y)$ and the same bounds $θ, M_{a}$ , with essential extrema $1, - 1$ on $B_{1}$ . Oscillations scale by $L$ , so it remains to prove ${ess osc}_{B_{1 / 2}} v \leq 2 - η_{0}$ for a universal $η_{0} > 0$ . The dichotomy gives the required half-level measure bound on $B_{1}$ .

1.2givenF3F4F5algebra

The measure estimate. Let $w \in H^{1} (B_{R})$ and $t < T$ . Then $∣ {w \leq t} \cap B_{R} ∣ ∣ {w \geq T} \cap B_{R} ∣^{1 - 1 / n} \leq \frac{C_{n}}{T - t} ∣ B_{R} ∣ \int_{B_{R}} ∣ D w ∣ d x$ . For smooth $w$ , fix $y$ with $w (y) \geq T$ and write $x = y + r ω$ for $x$ with $w (x) \leq t$ . Along the segment, $T - t \leq w (y) - w (x) \leq \int_{0}^{r} ∣ D w (y + s ω) ∣ d s$ . Integrating over the low set in polar coordinates, interchanging the radial integrals, and using $r \leq 2 R$ gives $∣ {w \leq t} \cap B_{R} ∣ \leq \frac{C_{n} ∣ B_{R} ∣}{T - t} \int_{B_{R}} \frac{∣ D w (z) ∣}{∣ z - y ∣^{n - 1}} d z .$ Integrate this in $y$ over $H : = {w \geq T} \cap B_{R}$ . For every measurable $E$ of finite measure, splitting the kernel integral at radius $∣ E ∣^{1 / n}$ gives $\sup_{z} \int_{E} ∣ z - y ∣^{1 - n} d y \leq C_{n} ∣ E ∣^{1 / n}$ ; hence the asserted inequality follows after division by $∣ H ∣^{1 / n}$ (the cases $∣ H ∣ = 0$ or $∣ {w \leq t} ∣ = 0$ are immediate). For general $w$ , choose $w_{j} \in C^{\infty} ({\overline{B}}_{R})$ converging strongly in $H^{1}$ and a subsequence converging a.e. Given $0 < ϵ < (T - t) / 2$ , apply the smooth inequality to $w_{j}$ at levels $t + ϵ, T - ϵ$ . Pointwise lower limits of the indicators dominate those of ${w \leq t}$ and ${w \geq T}$ ; [F5] passes the left side to the limit, while strong $L^{2}$ convergence of gradients gives convergence of $\int ∣ D w_{j} ∣$ . Letting $ϵ ↓ 0$ proves the claim. For its transition-set form, apply it to $z : = \min {(w - t)^{+}, T - t}$ at levels $0, T - t$ . Then ${z \leq 0} = {w \leq t}$ , ${z \geq T - t} = {w \geq T}$ , and $D z = 1_{{t < w < T}} D w$ a.e. Thus if $∣ {w \leq t} \cap B_{R} ∣ \geq γ ∣ B_{R} ∣$ , then Cauchy--Schwarz gives $∣ {w \geq T} \cap B_{R} ∣^{1 - 1 / n} \leq \frac{C_{n}}{γ (T - t)} ∣ {t < w < T} \cap B_{R} ∣^{1 / 2} (\int_{B_{R}} ∣ D (w - t)^{+} ∣^{2})^{1 / 2} .$ The Sobolev truncation chain rule also gives $D w = 0$ a.e. on the endpoint level sets.

2.1step 1.2F2algebra

The telescoping iteration. Work with the normalised $v$ of step 1.1 on $B_{1}$ and suppose first $∣ {v > 0} \cap B_{1} ∣ \leq \frac{1}{2} ∣ B_{1} ∣$ . Set $s : = (3 / 4)^{1 / n}$ , so $∣ B_{s} ∣ = \frac{3}{4} ∣ B_{1} ∣$ , and put $T_{k} : = 1 - 2^{- k - 1}$ , $E_{k} : = {v > T_{k}} \cap B_{s}$ , and $M_{k} : = ∣ E_{k} ∣$ . Since ${v \leq 0} \cap B_{1} = B_{1} ∖ ({v > 0} \cap B_{1})$ has measure at least $\frac{1}{2} ∣ B_{1} ∣$ and $T_{k} > 0$ , it follows that $∣ {v \leq T_{k}} \cap B_{s} ∣ \geq \frac{1}{2} ∣ B_{1} ∣ - ∣ B_{1} ∖ B_{s} ∣ = \frac{1}{4} ∣ B_{1} ∣ = \frac{1}{3} ∣ B_{s} ∣$ for every $k$ . The truncated Caccioppoli estimate of [F2], applied with outer radius $1$ and inner radius $s$ , gives $\int_{B_{s}} ∣ D (v - T_{k})^{+} ∣^{2} \leq C (n, θ, M_{a}) (1 - T_{k})^{2} ∣ B_{1} ∣$ , since $v \leq 1$ a.e. on $B_{1}$ . Apply the transition-set inequality of step 1.2 on $B_{s}$ with $t = T_{k}$ , $T = T_{k + 1}$ , and $γ = 1 / 3$ . As $T_{k + 1} - T_{k} = (1 - T_{k}) / 2$ , the level gap cancels the Caccioppoli factor and yields $M_{k + 1}^{1 - 1 / n} \leq C (n, θ, M_{a}) ∣ B_{1} ∣^{1 / 2} (M_{k} - M_{k + 1})^{1 / 2}, M_{k + 1}^{2 - 2 / n} \leq C (n, θ, M_{a}) (M_{k} - M_{k + 1}) .$ The constant absorbs the fixed volume $∣ B_{1} ∣$ .

3.1step 2.1algebra

Summation. Summing the inequalities of step 2.1 over $k = 0, \dots, N - 1$ and using $M_{k + 1} \geq M_{N}$ gives $N M_{N}^{2 - 2 / n} \leq C \sum_{k = 0}^{N - 1} (M_{k} - M_{k + 1}) = C (M_{0} - M_{N}) \leq C ∣ B_{1} ∣$ , hence $M_{N} \leq C (n, θ, M_{a}) N^{- n / (2 n - 2)} ∣ B_{1} ∣$ for every $N \geq 1$ .

4.1step 3.1F1F2algebra

The top level set is finally small. By [F2], $v_{N} : = (v - T_{N})^{+}$ is a nonnegative subsolution of $L_{0} w = 0$ . Apply the local boundedness estimate [F1] on outer ball $B_{s}$ with inner ratio $(2 s)^{- 1}$ ; since $B_{1 / 2} \subset B_{s}$ and $\int_{B_{s}} v_{N}^{2} \leq M_{N} (1 - T_{N})^{2}$ , this gives ${ess sup}_{B_{1 / 2}} v_{N} \leq C_{1} (n, θ, M_{a}) (∣ B_{s} ∣^{- 1} M_{N})^{1 / 2} (1 - T_{N}) \leq C_{2} N^{- n / (4 n - 4)} (1 - T_{N})$ by step 3.1. Choose $N = N (n, θ, M_{a}) \geq 1$ so large that $C_{2} N^{- n / (4 n - 4)} \leq \frac{1}{2}$ ; then $v \leq T_{N} + \frac{1}{2} (1 - T_{N}) = 1 - η_{0}$ on $B_{1 / 2}$ with $η_{0} : = \frac{1}{2} (1 - T_{N}) > 0$ .

5.1step 1.1step 4.1algebra∎

Conclusion of the reduction. If instead $∣ {v < 0} \cap B_{1} ∣ \leq \frac{1}{2} ∣ B_{1} ∣$ , steps 2.1-4.1 apply verbatim to $- v$ (which is again a solution of the homogeneous equation) and give $v \geq - 1 + η_{0}$ on $B_{1 / 2}$ . In the first case ${ess sup}_{B_{1 / 2}} v \leq 1 - η_{0}$ and ${ess inf}_{B_{1 / 2}} v \geq - 1$ , in the second ${ess sup}_{B_{1 / 2}} v \leq 1$ and ${ess inf}_{B_{1 / 2}} v \geq - 1 + η_{0}$ ; in both cases ${ess osc}_{B_{1 / 2}} v \leq 2 - η_{0}$ . Undoing the affine normalisation of step 1.1 multiplies both oscillations by $L$ and preserves the radius ratio, so ${ess osc}_{B_{R / 2} (x_{0})} u \leq (1 - η_{0} / 2) {ess osc}_{B_{R} (x_{0})} u$ , which is claim 2 with $η : = 1 - η_{0} / 2 \in (0, 1)$ and with $η_{0}$ , hence $η$ , depending only on $n, θ, M_{a}$ .

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

De Giorgi-Nash interior Holder regularity for divergence-form equations

Statement

Assume Countable Choice and the Axiom of Choice. Let $n \geq 2$ , let $Ω \subseteq R^{n}$ be open, and let $A$ , $L_{0}$ be as in De Giorgi local boundedness of homogeneous subsolutions, with measurable symmetric uniformly elliptic coefficients and constants $θ, M_{a}$ . Let $u \in H^{1} (Ω; R)$ be a weak solution of $L_{0} u = 0$ on $Ω$ . Then there are $α = α (n, θ, M_{a}) \in (0, 1)$ and, for every $α^{'} \in (0, α)$ , a class $u^{*} \in C_{l o c}^{0, α^{'}} (Ω)$ (Local Hölder and scaled C-two-alpha norms on balls, Hölder spaces $C^{k, α}$ , closure and interior scaled norms, and $C^{k, α}$ domains) with $u^{*} = u$ a.e. on $Ω$ , and for every ball $B_{R_{0}} (x_{0}) ⋐ Ω$ , $[u^{*}]_{0, α^{'}; B_{R_{0} / 2} (x_{0})} \leq C (n, θ, M_{a}, α^{'}) R_{0}^{- α^{'}} (\frac{1}{∣ B_{R_{0}} (x_{0}) ∣} \int_{B_{R_{0}} (x_{0})} u^{2} d x)^{1 / 2},$ and $∥ u^{*} ∥_{L^{\infty} (B_{R_{0} / 2} (x_{0}))} \leq C R_{0}^{- n / 2} ∥ u ∥_{L^{2} (B_{R_{0}} (x_{0}))}$ . In particular every real weak solution of the homogeneous scalar equation with the symmetric bounded measurable uniformly elliptic principal coefficients specified above has a locally Holder continuous representative, and the representative is unique up to equality everywhere on $Ω$ .

Facts & Assumptions

Given: Countable Choice and the Axiom of Choice; an open $Ω \subseteq R^{n}$ , $n \geq 2$ ; measurable symmetric uniformly elliptic coefficients $A$ with constants $θ, M_{a}$ ; the principal operator $L_{0} u = - D_{i} (a^{i j} D_{j} u)$ with form $a_{0}$ ; a real weak solution $u \in H^{1} (Ω; R)$ ; and a ball $B_{R_{0}} (x_{0}) ⋐ Ω$ .

[F1]

One-step oscillation reduction: there is $η = η (n, θ, M_{a}) \in (0, 1)$ such that for each ball $B_{R} (x)$ with $B_{2 R} (x) ⋐ Ω$ , ${ess osc}_{B_{R / 2} (x)} u \leq η {ess osc}_{B_{R} (x)} u$ (De Giorgi oscillation reduction: one half-level set is small).

[F2]

Local boundedness for a nonnegative subsolution: for every nonnegative weak subsolution $w$ of $L_{0} w = 0$ and every ball $B_{R} (x) ⋐ Ω$ , $0 < ρ < 1$ and $p > 0$ , ${ess sup}_{B_{ρ R} (x)} w \leq C (n, θ, M_{a}, ρ, p) (\frac{1}{∣ B_{R} (x) ∣} \int_{B_{R} (x)} w^{p})^{1 / p}$ (De Giorgi local boundedness of homogeneous subsolutions).

[F3]

Extend $u \in L^{2} (Ω)$ by zero off $Ω$ . The extension lies in $L^{2} (R^{n})$ and hence $L_{l o c}^{1} (R^{n})$ by Holder on bounded sets. The cited Lebesgue-point theorem applies to this extension; restriction back to $Ω$ gives a full-measure Lebesgue set, dense because every nonempty open subset has positive measure (Almost every point is a Lebesgue point of a locally integrable function, Lebesgue points and the Lebesgue set of an $L_{l o c}^{1}$ class, The average of a locally integrable function over a Euclidean ball).

[F4]

The target $R$ is complete by The reals are complete. Apply the dense-set extension theorem on each smaller ball, where the local Holder bound gives uniform continuity; the extensions agree on overlaps because they agree on the dense Lebesgue set. This gives a unique continuous extension on the ambient open set and passes the local Holder bounds to it (A uniformly continuous map from a dense subspace into a complete metric space extends uniquely to a uniformly continuous map on the whole space, Complete metric space: every Cauchy sequence converges in the space).

[F5]

For continuous functions, pointwise supremum and infimum on an open ball equal the essential supremum and infimum of the corresponding almost-everywhere class; the Holder seminorm and norm are those of Local Hölder and scaled C-two-alpha norms on balls and Hölder spaces $C^{k, α}$ , closure and interior scaled norms, and $C^{k, α}$ domains (The essential supremum of a measurable function with respect to a measure).

[F6]

Positive parts of a real weak solution of the homogeneous equation are weak subsolutions. For $v = u$ or $v = - u$ , the zero-source identity extends from $C_{c}^{\infty}$ tests to $H_{0}^{1}$ by density and boundedness of the form (Zero-boundary Sobolev space as a norm closure, The elliptic form is well defined and bounded on $H^{1}$ ). Thus test with the nonnegative $H_{0}^{1}$ function $ϕ χ_{ϵ} (v)$ , where $ϕ \in C_{c}^{\infty} (Ω)$ is nonnegative and $χ_{ϵ} (t) = \min {1, t^{+} / ϵ}$ . The chain and product rules give $a_{0} (v, ϕ χ_{ϵ} (v)) = \int χ_{ϵ} (v) A D v \cdot D ϕ + \int ϕ χ_{ϵ}^{'} (v) A D v \cdot D v = 0$ ; the second term is nonnegative. Dominated convergence in the first term as $ϵ ↓ 0$ gives $a_{0} (v^{+}, ϕ) \leq 0$ (Chain rule for globally Lipschitz scalar maps of Sobolev functions, Weak Leibniz rule with a smooth factor, Positive-part truncation calculus and admissible cut-off weak tests).

Proof

technique · iterate the one-step oscillation reduction on smaller interior balls, transfer its dyadic decay to Lebesgue values, extend those values continuously, and use local boundedness of the positive and negative parts for the quantitative norm estimate

1.1givenF2F6

Local boundedness of the positive and negative parts. For $v = u$ and $v = - u$ , [F6] shows that $v^{+}$ is a nonnegative weak subsolution. Given any ball $B_{S} (x) ⋐ Ω$ , choose $S^{'} > S$ with $B_{S^{'}} (x) ⋐ Ω$ and apply [F2] on $B_{S^{'}} (x)$ with inner ratio $S / S^{'}$ and exponent $p = 2$ . Since $v^{+} \in H^{1} (Ω)$ , its $L^{2} (B_{S^{'}})$ mean is finite, so both $u^{+}$ and $(- u)^{+}$ are essentially bounded on $B_{S} (x)$ . Consequently $u$ has finite essential oscillation on every compactly contained ball.

2.1step 1.1F1algebra

Geometric oscillation decay. Fix $B_{R} (x_{0}) ⋐ Ω$ . By [F1], applying the one-step estimate first with outer ball $B_{R / 2}$ and then with successive dyadic outer balls gives ${ess osc}_{B_{R / 2^{k + 1}} (x_{0})} u \leq η^{k} {ess osc}_{B_{R} (x_{0})} u$ for every integer $k \geq 0$ . By monotonicity of essential oscillation, if $0 < r \leq R / 4$ , choosing $k$ so that $R / 2^{k + 2} < r \leq R / 2^{k + 1}$ gives ${ess osc}_{B_{r} (x_{0})} u \leq C_{0} (r / R)^{α_{0}} {ess osc}_{B_{R} (x_{0})} u, α_{0} : = \frac{\log (1 / η)}{\log 2} > 0,$ with $C_{0} = 4^{α_{0}}$ ; for $R / 4 < r \leq R$ the same inequality follows from monotonicity and this choice of $C_{0}$ . This argument applies to any ball compactly contained in $Ω$ , and all oscillations are finite by step 1.1.

3.1step 2.1F3algebra

Holder modulus at Lebesgue points. Fix $0 < ρ < 1$ and $x, y \in B_{ρ R} (x_{0})$ that are Lebesgue points of $u$ . Put $m : = (1 - ρ) R / 2$ and $d : = ∣ x - y ∣$ . If $0 < d < m / 4$ , then $B_{2 d} (x) \subset B_{2 m} (x) ⋐ B_{R} (x_{0})$ . The decay of step 2.1 applied to $B_{2 m} (x)$ gives ${ess osc}_{B_{2 d} (x)} u \leq C_{0} (d / m)^{α_{0}} {ess osc}_{B_{R} (x_{0})} u .$ For sufficiently small $s > 0$ , both $B_{s} (x)$ and $B_{s} (y)$ lie in $B_{2 d} (x)$ ; their averages lie between its essential infimum and supremum. Passing to the Lebesgue limits gives $∣ u^{*} (x) - u^{*} (y) ∣ \leq {ess osc}_{B_{2 d} (x)} u$ . If instead $d \geq m / 4$ , the bound $∣ u^{*} (x) - u^{*} (y) ∣ \leq {ess osc}_{B_{R} (x_{0})} u$ suffices. In either case, $∣ u^{*} (x) - u^{*} (y) ∣ \leq C_{ρ} (d / R)^{α_{0}} {ess osc}_{B_{R} (x_{0})} u,$ where $C_{ρ}$ depends only on $η, ρ$ .

4.1step 3.1F3F4

The continuous representative. The Lebesgue set of $u$ is dense by [F3]. Step 3.1 makes the Lebesgue representative locally Holder on its intersection with each smaller ball $B_{ρ R} (x_{0})$ . The extension theorem [F4] gives a unique continuous extension on $Ω$ , still denoted $u^{*}$ , which agrees with $u$ a.e. and retains these local Holder bounds.

5.1step 3.1step 4.1F2F5algebra

Holder and supremum estimates. Let $R : = R_{0}$ and apply step 3.1 on the outer ball $B_{3 R / 4} (x_{0})$ with inner ratio $2 / 3$ . Applying [F2] with $p = 2$ and outer ball $B_{R}$ to the positive parts $u^{+}$ and $(- u)^{+}$ from step 1.1 gives ${ess sup}_{B_{3 R / 4}} ∣ u ∣ \leq C (\frac{1}{∣ B_{R} ∣} \int_{B_{R}} u^{2})^{1 / 2} .$ Hence ${ess osc}_{B_{3 R / 4}} u \leq 2 C (\frac{1}{∣ B_{R} ∣} \int_{B_{R}} u^{2})^{1 / 2}$ . Steps 3.1 and 4.1 give the corresponding increment bound with exponent $α_{0}$ . Set $α : = \min {α_{0}, 1 / 2} \in (0, 1)$ ; weakening the exponent to $α$ preserves the estimate. For $0 < α^{'} < α$ , interpolate that Holder increment with the supremum bound: $\min {C M (∣ x - y ∣ / R)^{α}, 2 M} \leq C^{'} (α^{'}) M (∣ x - y ∣ / R)^{α^{'}}$ , where $M = (\frac{1}{∣ B_{R} ∣} \int_{B_{R}} u^{2})^{1 / 2}$ . Thus $[u^{*}]_{0, α^{'}; B_{R / 2}} \leq C R^{- α^{'}} (\frac{1}{∣ B_{R} ∣} \int_{B_{R}} u^{2})^{1 / 2}, ∥ u^{*} ∥_{L^{\infty} (B_{R / 2})} \leq C R^{- n / 2} ∥ u ∥_{L^{2} (B_{R})} .$ The constants depend only on $n, θ, M_{a}, α^{'}$ .

6.1F3F4∎

Uniqueness. If two continuous representatives agree with $u$ a.e., they agree on a full-measure, hence dense, subset of $Ω$ ; continuity makes them equal everywhere. All arguments use only the declared choice principles.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

Geometric oscillation decay implies a Hölder modulus

Statement

Let $n \geq 1$ , let $Ω \subseteq R^{n}$ be open, let $u : Ω \to R$ and let $θ \in (0, 1)$ satisfy ${osc}_{B_{r} (x)} u \leq θ {osc}_{B_{2 r} (x)} u whenever B_{2 r} (x) ⋐ Ω,$ where ${osc}_{B} u : = \sup_{B} u - \inf_{B} u$ , and assume ${osc}_{B_{R} (x_{0})} u < \infty$ for every $B_{R} (x_{0}) ⋐ Ω$ . Put $α_{0} : = \frac{\log (1 / θ)}{\log 2} > 0$ and $α : = \min {α_{0}, 1 / 2} \in (0, 1)$ . Then for every ball $B_{R} (x_{0}) ⋐ Ω$ and all $x, y \in B_{R / 2} (x_{0})$ , $∣ u (x) - u (y) ∣ \leq 4^{α} (\frac{∣ x - y ∣}{R})^{α} {osc}_{B_{R} (x_{0})} u,$ so $u$ is locally $α$ -Hölder in $Ω$ (Local Hölder and scaled C-two-alpha norms on balls) with $[u]_{0, α; B_{R / 2} (x_{0})} \leq 4^{α} R^{- α} {osc}_{B_{R} (x_{0})} u$ ; moreover for every $0 < α^{'} < α$ the same estimate holds with $α^{'}$ and constant $4^{α^{'}}$ .

Facts & Assumptions

Given: an integer $n \geq 1$ , an open set $Ω \subseteq R^{n}$ , a function $u : Ω \to R$ with finite oscillation on every compactly contained ball, a number $θ \in (0, 1)$ with ${osc}_{B_{r} (x)} u \leq θ {osc}_{B_{2 r} (x)} u$ whenever $B_{2 r} (x) ⋐ Ω$ , and $α_{0} = \log (1 / θ) / \log 2$ , $α = \min {α_{0}, 1 / 2}$ .

[F1]

For every $x$ and $r > 0$ , ${osc}_{B_{r} (x)} u = \sup_{B_{r} (x)} u - \inf_{B_{r} (x)} u \in [0, + \infty]$ , and if $A \subseteq B \subseteq Ω$ then ${osc}_{A} u \leq {osc}_{B} u$ , because a supremum over a smaller set is no larger and an infimum over a smaller set is no smaller.

[F2]

Since $2^{α_{0}} = 1 / θ$ and $θ = 2^{- α_{0}}$ by the definition of the real power, for $0 < a \leq 1$ the map $β \mapsto a^{β}$ is nonincreasing, and for $a, b > 0$ and real $β$ one has $(a b)^{β} = a^{β} b^{β}$ (Real powers for positive bases, with the zero-base positive-exponent convention).

[F3]

For a ball $B_{R} (x_{0})$ , the seminorm $[u]_{0, α; B_{R} (x_{0})}$ is the supremum of $∣ u (x) - u (y) ∣ / ∣ x - y ∣^{α}$ over all $x, y \in B_{R} (x_{0})$ with $x \neq y$ (Local Hölder and scaled C-two-alpha norms on balls).

Proof

technique · direct dyadic iteration of the oscillation hypothesis

1.1givenF1

Fix a ball $B_{R} (x_{0}) ⋐ Ω$ . The claim is immediate when $x = y$ , so assume $x \neq y$ and put $d : = ∣ x - y ∣ > 0$ and $z : = (x + y) / 2$ . Since $x, y \in B_{R / 2} (x_{0})$ , the midpoint satisfies $z \in B_{R / 2} (x_{0})$ and $d < R$ . The oscillation of $u$ on $B_{R} (x_{0})$ is finite by hypothesis. If $d \geq R / 2$ , then $∣ u (x) - u (y) ∣ \leq {osc}_{B_{R} (x_{0})} u \leq 4^{α} (d / R)^{α} {osc}_{B_{R} (x_{0})} u$ , so assume henceforth $d < R / 2$ .

1.2givenF1algebra

Let $k \geq 0$ be the largest integer with $2^{k + 1} d \leq R$ ; it exists because $d < R / 2$ , and the set of admissible exponents is bounded above. For every $0 \leq j \leq k$ one has $B_{2^{j} d} (z) ⋐ B_{R} (x_{0})$ : the midpoint $z$ is within $R / 2$ of $x_{0}$ , while $2^{j} d \leq R / 2$ . Consequently the given oscillation hypothesis applies to the pair of radii $2^{j - 1} d$ and $2^{j} d$ for every $1 \leq j \leq k$ .

2.1step 1.2F1

Iterating the hypothesis, ${osc}_{B_{d} (z)} u \leq θ^{k} {osc}_{B_{2^{k} d} (z)} u$ . Indeed the case $k = 0$ is an equality, and if the claim holds for $k - 1$ then it holds for $k$ by appending the one step ${osc}_{B_{2^{k - 1} d} (z)} u \leq θ {osc}_{B_{2^{k} d} (z)} u$ supplied by step 1.2. Since $B_{2^{k} d} (z) \subseteq B_{R} (x_{0})$ , monotonicity of the oscillation gives ${osc}_{B_{d} (z)} u \leq θ^{k} {osc}_{B_{R} (x_{0})} u$ .

2.2step 1.2F2algebra

By maximality of $k$ , $2^{k + 2} d > R$ , so $2^{- k} < 4 d / R$ . Since $2^{- k} \leq 1$ and $α \leq α_{0}$ , [F2] gives $θ^{k} = (2^{- k})^{α_{0}} \leq (2^{- k})^{α} < (4 d / R)^{α} = 4^{α} (d / R)^{α}$ .

3.1step 2.1step 2.2F1F3

Since $x, y \in B_{d} (z)$ , combining steps 2.1 and 2.2 gives $∣ u (x) - u (y) ∣ \leq {osc}_{B_{d} (z)} u \leq 4^{α} (d / R)^{α} {osc}_{B_{R} (x_{0})} u$ , which is the displayed inequality because $d = ∣ x - y ∣$ . Dividing by $∣ x - y ∣^{α}$ and taking the supremum over $x \neq y$ in $B_{R / 2} (x_{0})$ yields $[u]_{0, α; B_{R / 2} (x_{0})} \leq 4^{α} R^{- α} {osc}_{B_{R} (x_{0})} u$ by [F3]; since every point of $Ω$ has a ball $B_{R} (x_{0}) ⋐ Ω$ about it and $R / 2$ is available, $u$ is locally $α$ -Hölder on $Ω$ .

4.1step 2.1step 2.2F2given∎

For the exponent clause, fix $0 < α^{'} < α$ . If $d < R / 4$ , then $4 d / R < 1$ , and $α^{'} < α \leq α_{0}$ gives $θ^{k} = (2^{- k})^{α_{0}} \leq (2^{- k})^{α^{'}} < (4 d / R)^{α^{'}}$ . If $d \geq R / 4$ , then $∣ u (x) - u (y) ∣ \leq {osc}_{B_{R} (x_{0})} u \leq 4^{α^{'}} (d / R)^{α^{'}} {osc}_{B_{R} (x_{0})} u$ . Combining these cases proves the claimed $α^{'}$ estimate with constant $4^{α^{'}}$ ; no choice principle is used.

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Logarithmic Caccioppoli estimate for positive supersolutions

Statement

Assume Countable Choice and the Axiom of Choice. Let $n \geq 2$ , let $Ω \subseteq R^{n}$ be open, let $A$ and $L_{0}$ be as in De Giorgi local boundedness of homogeneous subsolutions, and let $u \in H^{1} (Ω; R)$ satisfy $u > 0$ a.e. on $Ω$ and $a_{0} (u, v) \geq 0 for every v \in H_{0}^{1} (Ω), v \geq 0 a . e .,$ i.e. $u$ is a positive weak supersolution of $L_{0} u = 0$ (Weak subsolutions and supersolutions of a divergence-form equation). Then for every $η \in C_{c}^{\infty} (Ω)$ and every $ε > 0$ , $\int_{Ω} η^{2} ∣ D \log (u + ε) ∣^{2} d x \leq \frac{4 M_{a}^{2}}{θ} \int_{Ω} ∣ D η ∣^{2} d x,$ and consequently, for concentric balls $B_{r} (x_{0}) ⋐ B_{R} (x_{0}) ⋐ Ω$ , $\int_{B_{r} (x_{0})} \frac{∣ D u ∣^{2}}{u^{2}} d x \leq C (n, θ, M_{a}) \frac{∣ B_{R} (x_{0}) ∣}{(R - r)^{2}},$ the second inequality being the monotone limit $ε ↓ 0$ of the first. No lower bound on $u$ is assumed away from its positivity.

Facts & Assumptions

Given: Countable Choice and the Axiom of Choice; an open $Ω \subseteq R^{n}$ , $n \geq 2$ ; a measurable symmetric coefficient field $A$ with $θ ∣ ξ ∣^{2} \leq ⟨ A ξ, ξ ⟩ \leq M_{a}^{2} ∣ ξ ∣^{2}$ a.e.; a class $u \in H^{1} (Ω; R)$ with $u > 0$ a.e. and $a_{0} (u, v) \geq 0$ for every nonnegative $v \in H_{0}^{1} (Ω)$ ; $η \in C_{c}^{\infty} (Ω)$ ; $ε > 0$ .

[F1]

Assume Countable Choice. The scalar map $F_{ε} (t) : = (\max {t, 0} + ε)^{- 1}$ is globally Lipschitz. Its composition with $u$ lies in $H_{l o c}^{1} (Ω)$ and, since $u > 0$ a.e., equals $(u + ε)^{- 1}$ with derivative $D ((u + ε)^{- 1}) = - (u + ε)^{- 2} D u$ (Chain rule for globally Lipschitz scalar maps of Sobolev functions, Integer-order Sobolev spaces and their norms).

[F2]

Assume Countable Choice. Products with smooth compactly supported cutoffs: $η^{2} w_{ε} \in H_{0}^{1} (Ω)$ with $D (η^{2} w_{ε}) = 2 η w_{ε} D η + η^{2} D w_{ε}$ , because $η w_{ε}$ is compactly supported and lies in $H_{0}^{1}$ (Weak Leibniz rule with a smooth factor, Positive-part truncation calculus and admissible cut-off weak tests).

[F3]

Matrix Cauchy-Schwarz and Young: for the positive definite field $A$ , $∣ a^{i j} ξ_{j} ζ_{i} ∣ \leq (a^{i j} ξ_{j} ξ_{i})^{1 / 2} (a^{i j} ζ_{i} ζ_{j})^{1 / 2} \leq M_{a} ∣ ζ ∣ (a^{i j} ξ_{j} ξ_{i})^{1 / 2}$ ; and $2 M_{a} X Y \leq \frac{1}{2} θ X^{2} + \frac{2 M_{a}^{2}}{θ} Y^{2}$ for $X, Y \geq 0$ , $θ > 0$ (Young's inequality for conjugate real exponents, Holder's inequality for integrals, including the endpoint cases).

[F4]

Bumps: for $0 < r < R$ there is $η \in C_{c}^{\infty} (B_{R} (x_{0}))$ with $η = 1$ on $B_{r} (x_{0})$ and $∣ D η ∣ \leq C_{U} / (R - r)$ for a universal $C_{U}$ ; the explicit radial construction gives this bound (A smooth bump between concentric Euclidean balls, Compactly supported scaled Euclidean bumps).

Proof

technique · direct; insert the regularised reciprocal test function, expand, and absorb the cross term by Young's inequality with the ellipticity constant

1.1givenF1F2

The test function and the supersolution inequality. By [F1] and [F2], $v_{ε} : = η^{2} (u + ε)^{- 1}$ is a nonnegative element of $H_{0}^{1} (Ω)$ with $D_{i} v_{ε} = 2 η (u + ε)^{- 1} D_{i} η - η^{2} (u + ε)^{- 2} D_{i} u$ . Testing the supersolution inequality with $v_{ε}$ gives $0 \leq a_{0} (u, v_{ε}) = 2 \int_{Ω} η (u + ε)^{- 1} a^{i j} D_{j} u D_{i} η d x - \int_{Ω} η^{2} (u + ε)^{- 2} a^{i j} D_{j} u D_{i} u d x$ . Taking absolute values in the cross term yields $\int_{Ω} η^{2} (u + ε)^{- 2} a^{i j} D_{j} u D_{i} u d x \leq 2 \int_{Ω} ∣ η ∣ (u + ε)^{- 1} ∣ a^{i j} D_{j} u D_{i} η ∣ d x$ , which is valid even when the allowed cutoff $η$ changes sign.

2.1step 1.1F3algebra

Ellipticity and absorption. Write $C : = (\int_{Ω} η^{2} (u + ε)^{- 2} a^{i j} D_{j} u D_{i} u d x)^{1 / 2}$ and $B : = (\int_{Ω} ∣ D η ∣^{2} d x)^{1 / 2}$ . By step 1.1 and [F3], $C^{2} \leq 2 M_{a} B C$ , so $C \leq 2 M_{a} B$ if $C > 0$ (and the same bound is trivial otherwise). Since also $θ \int_{Ω} η^{2} ∣ D \log (u + ε) ∣^{2} d x = θ \int_{Ω} η^{2} (u + ε)^{- 2} ∣ D u ∣^{2} d x \leq C^{2}$ , we obtain $\int_{Ω} η^{2} ∣ D \log (u + ε) ∣^{2} d x \leq \frac{4 M_{a}^{2}}{θ} \int_{Ω} ∣ D η ∣^{2} d x$ , the first displayed estimate.

3.1step 2.1F4algebra∎

The ball form. Let $0 < r < R$ with $B_{R} (x_{0}) ⋐ Ω$ and choose the bump $η$ of [F4]; then $\int_{Ω} η^{2} ∣ D \log (u + ε) ∣^{2} d x \geq \int_{B_{r} (x_{0})} ∣ D u ∣^{2} (u + ε)^{- 2} d x$ and $\int_{Ω} ∣ D η ∣^{2} d x \leq C_{U}^{2} ∣ B_{R} (x_{0}) ∣ (R - r)^{- 2}$ , so $\int_{B_{r} (x_{0})} ∣ D u ∣^{2} (u + ε)^{- 2} d x \leq 4 C_{U}^{2} M_{a}^{2} θ^{- 1} ∣ B_{R} (x_{0}) ∣ (R - r)^{- 2}$ . Since $(u + ε)^{- 2} ↑ u^{- 2}$ as $ε ↓ 0$ , the monotone convergence theorem applied to the nonnegative integrands $∣ D u ∣^{2} (u + ε)^{- 2}$ yields $\int_{B_{r} (x_{0})} ∣ D u ∣^{2} u^{- 2} d x \leq C (n, θ, M_{a}) ∣ B_{R} (x_{0}) ∣ (R - r)^{- 2}$ with $C (n, θ, M_{a}) : = 4 C_{U}^{2} M_{a}^{2} / θ$ ; no lower bound on $u$ is used beyond positivity, and only the declared choice principles are used.

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Moser iteration for positive supersolutions: negative-power and logarithmic comparison

Statement

Assume Countable Choice and the Axiom of Choice. Let $n \geq 2$ , let $Ω \subseteq R^{n}$ be open, let $A$ and $L_{0}$ be as in De Giorgi local boundedness of homogeneous subsolutions, and let $u \in H^{1} (Ω; R)$ with $u > 0$ a.e. be a positive weak supersolution of $L_{0} u = 0$ . Then the negative-power chain of the Moser iteration holds: for every $0 < ρ < 1$ , every $p > 0$ and every ball $B_{R} (x_{0}) ⋐ Ω$ , $(\frac{1}{∣ B_{R} (x_{0}) ∣} \int_{B_{R} (x_{0})} u^{- p} d x)^{- 1 / p} \leq C (n, θ, M_{a}, ρ, p) {ess inf}_{B_{ρ R} (x_{0})} u .$ Moreover there is an exponent $p_{0} = p_{0} (n, θ, M_{a}) > 0$ such that, for every $B_{R} (x_{0}) ⋐ Ω$ , $(\frac{1}{∣ B_{3 R / 4} (x_{0}) ∣} \int_{B_{3 R / 4} (x_{0})} u^{p_{0}} d x)^{1 / p_{0}} \leq C (n, θ, M_{a}) (\frac{1}{∣ B_{3 R / 4} (x_{0}) ∣} \int_{B_{3 R / 4} (x_{0})} u^{- p_{0}} d x)^{- 1 / p_{0}} .$ The proof does not assume that $u$ is bounded away from zero: the negative-power tests and positive moments are handled after regularisation $u \mapsto u + ε$ ; monotone convergence passes the increasing negative moments, while dominated convergence passes the decreasing positive moments. The logarithmic estimate Logarithmic Caccioppoli estimate for positive supersolutions supplies the input for the comparison of opposite powers.

Facts & Assumptions

Given: Countable Choice and the Axiom of Choice; an open set $Ω \subseteq R^{n}$ , $n \geq 2$ ; measurable symmetric coefficients $A$ with $θ ∣ ξ ∣^{2} \leq ⟨ A (x) ξ, ξ ⟩ \leq M_{a}^{2} ∣ ξ ∣^{2}$ ; the principal operator $L_{0} u = - D_{i} (a^{i j} D_{j} u)$ and its form $a_{0}$ ; a class $u \in H^{1} (Ω; R)$ with $u > 0$ a.e. and $a_{0} (u, v) \geq 0$ for every nonnegative $v \in H_{0}^{1} (Ω)$ ; a ball $B_{R} (x_{0}) ⋐ Ω$ and $0 < ρ < 1$ .

[F1]

Assume the Axiom of Choice. Composition, products and density: a globally Lipschitz scalar composition of an $H^{1}$ class obeys the Sobolev chain rule; a compactly supported smooth factor obeys the weak product rule; and a compactly supported $H^{1}$ class lies in $H_{0}^{1}$ by zero extension and smooth approximation. Thus the bounded truncation of $U^{β}$ and its cutoff test in step 1.1 are admissible (Chain rule for globally Lipschitz scalar maps of Sobolev functions, Weak Leibniz rule with a smooth factor, Compactly supported Sobolev functions extend by zero in every integer order, Compactly supported smooth functions are dense in W^{k,p}(R^n), Zero-boundary Sobolev space as a norm closure, Weak subsolutions and supersolutions of a divergence-form equation, Integer-order Sobolev spaces and their norms).

[F2]

Assume the Axiom of Choice. Ellipticity and boundedness of the coefficients: $θ ∣ ξ ∣^{2} \leq a^{i j} ξ_{i} ξ_{j} \leq M_{a}^{2} ∣ ξ ∣^{2}$ for a.e. point and every $ξ$ (Uniformly elliptic divergence-form operators and their sesquilinear forms, The elliptic form is well defined and bounded on $H^{1}$ ).

[F3]

Assume the Axiom of Choice. Sobolev input: there is $κ > 1$ , namely $κ = n / (n - 2)$ for $n \geq 3$ and any fixed finite $κ > 1$ for $n = 2$ , and a constant $S$ with $∥ w ∥_{L^{2 κ} (B_{1})} \leq S ∥ D w ∥_{L^{2} (B_{1})}$ for every $w \in H_{0}^{1} (B_{1})$ (The Sobolev inequality for zero-boundary Sobolev closures on open sets, The critical Sobolev embedding into every finite $L^{q}$ ). In dimension two the gradient-only form follows directly from the zero-boundary supplier: set $r = 2 κ / (κ + 1) \in (1, 2)$ , so $r^{*} = 2 κ$ . The same smooth approximants and finite measure put $w$ in $W_{0}^{1, r} (B_{1})$ , and Holder gives $∥ w ∥_{2 κ} \leq C (2, r) ∥ D w ∥_{r} \leq C (2, r) ∣ B_{1} ∣^{1 / (2 κ)} ∥ D w ∥_{2}$ .

[F4]

Assume the Axiom of Choice. If $g \in L^{\infty} (B_{ρ})$ on a finite-measure ball, then $∥ g ∥_{L^{q} (B_{ρ})} \to {ess sup}_{B_{ρ}} ∣ g ∣$ as $q \to \infty$ ( $L^{p}$ norms converge to the essential supremum for essentially bounded $L^{r}$ functions, The space $L^{p} (μ)$ as the quotient by null functions, The essential supremum of a measurable function with respect to a measure).

[F5]

Assume the Axiom of Choice. Logarithmic Caccioppoli estimate: for every $η \in C_{c}^{\infty} (Ω)$ and every $ε > 0$ , $\int_{Ω} η^{2} ∣ D \log (u + ε) ∣^{2} d x \leq \frac{4 M_{a}^{2}}{θ} \int_{Ω} ∣ D η ∣^{2} d x$ (Logarithmic Caccioppoli estimate for positive supersolutions).

[F6]

Assume the Axiom of Choice. Poincare-Wirtinger inequality on balls, and the existence of smooth bumps between concentric balls with $∣ D η ∣ \leq C / (s - r)$ (Poincare inequality on a ball, A smooth bump between concentric Euclidean balls).

[F7]

Assume the Axiom of Choice. Monotone and dominated convergence for the integral, used to pass to the limit $ε ↓ 0$ in the regularised estimates (Monotone convergence for the integral, Dominated convergence, The essential supremum of a measurable function with respect to a measure, The average of a locally integrable function over a Euclidean ball).

[F8]

Dyadic differentiation and layer cake: for a locally integrable function, averages over shrinking dyadic subcubes containing $x$ converge to its Lebesgue value at almost every $x$ . To use the whole-space supplier on a fixed covering cube, first zero-extend its integrable restriction. At a Lebesgue point $x$ , enclose each containing cube of side $s$ in $B_{\sqrt{n} s} (x)$ ; the volume ratio is fixed, so the cube average of $∣ g - g (x) ∣$ tends to zero by Almost every point is a Lebesgue point of a locally integrable function. For $j > 0$ , $\int ∣ g ∣^{j} = j \int_{0}^{\infty} t^{j - 1} ∣ {∣ g ∣ > t} ∣ d t$ , with extended nonnegative values (Lebesgue differentiation theorem on $R^{n}$ , For 0 < p < infinity, the layer-cake formula computes the integral of |f|^p from the distribution function).

Proof

Here $avg g$ denotes the normalized integral of $g$ over the ball in the surrounding estimate.

Proof technique: direct; regularise by $u + ε$ , test with bounded truncations of negative powers to derive a positive-power Sobolev iteration for $1 / (u + ε)$ , and use the scale-invariant logarithmic Caccioppoli estimate to obtain local mean oscillation, a dyadic stopping estimate and exponential integrability of the logarithm.

1.1givenF1F2F6algebra

Scaling, bounded truncation, density and the energy estimate. Under $y = (x - x_{0}) / R$ , the principal divergence form and the weak supersolution inequality retain the same ellipticity bounds, while ball averages are invariant; it is enough to work on $B_{1}$ . Fix $ε > 0$ , put $U = u + ε$ , and for $p > 0$ choose $β = - p - 1 < - 1$ . Then $U \geq ε$ , so $U^{β}$ and its weak gradient are bounded by constants (depending on $p, ε$ ) times $1$ and $∣ D u ∣$ , respectively. More explicitly, for $N > ε^{β}$ the globally Lipschitz bounded truncation $P_{ε, N} (t) : = \min {(\max {t, 0} + ε)^{β}, N}$ satisfies $P_{ε, N} (u) = U^{β}$ a.e. The cutoff product $η^{2} P_{ε, N} (u)$ lies in $H_{0}^{1} (B_{1})$ by the chain and product rules and compact-support zero extension; approximate it in $H_{0}^{1}$ by nonnegative smooth tests and use continuity of the form to pass the supersolution inequality to this test. Testing with $η^{2} U^{β}$ gives $(p + 1) \int_{B_{1}} η^{2} U^{- p - 2} ⟨ A D U, D U ⟩ d x \leq 2 \int_{B_{1}} ∣ η ∣ U^{- p - 1} ∣ ⟨ A D U, D η ⟩ ∣ d x .$ By Cauchy--Schwarz in the $A$ -energy, $∣ ⟨ A D U, D η ⟩ ∣ \leq ⟨ A D U, D U ⟩^{1 / 2} ⟨ A D η, D η ⟩^{1 / 2} \leq M_{a} ∣ D η ∣ ⟨ A D U, D U ⟩^{1 / 2}$ . Absorbing the resulting energy square root yields $\int_{B_{1}} η^{2} U^{- p - 2} ⟨ A D U, D U ⟩ d x \leq \frac{4 M_{a}^{2}}{(p + 1)^{2}} \int_{B_{1}} U^{- p} ∣ D η ∣^{2} d x .$ Ellipticity and $D (U^{- p / 2}) = - (p / 2) U^{- p / 2 - 1} D U$ then give $\int η^{2} ∣ D (U^{- p / 2}) ∣^{2} \leq \frac{M_{a}^{2} p^{2}}{(p + 1)^{2} θ} \int U^{- p} ∣ D η ∣^{2} \leq \frac{M_{a}^{2}}{θ} \int U^{- p} ∣ D η ∣^{2}$ .

1.2givenF5F6algebra

Local logarithmic oscillation. Put $w = \log U$ and $ℓ = \frac{1}{∣ B_{7 / 8} ∣} \int_{B_{7 / 8}} w$ . The logarithmic Caccioppoli estimate [F5], with a smooth cutoff supported in $B_{1}$ and equal to one on $B_{7 / 8}$ , gives $\int_{B_{7 / 8}} ∣ D w ∣^{2} d x \leq C (n, θ, M_{a})$ ; Poincare [F6] therefore gives $∥ w - ℓ ∥_{L^{2} (B_{7 / 8})} \leq C$ . Cover $B_{3 / 4}$ by finitely many axis-parallel cubes $Q_{0}$ of a fixed side $s_{n} > 0$ so small that their closures lie in $B_{13 / 16}$ and every concentric ball below lies in $B_{1}$ . For each dyadic subcube $Q$ of side $s$ , let $B$ be the concentric ball of radius $\sqrt{n} s$ , which contains $Q$ . The logarithmic estimate with a smooth cutoff equal to one on $B$ and supported in the concentric ball of radius $2 \sqrt{n} s$ gives $\int_{B} ∣ D w ∣^{2} \leq C s^{n - 2}$ . The ball Poincare inequality [F6] on $B$ , together with $∣ B ∣ / ∣ Q ∣ = ∣ B_{\sqrt{n}} ∣$ , then yields $\frac{1}{∣ Q ∣} \int_{Q} ∣ w - w_{Q} ∣ \leq K$ , with $K = K (n, θ, M_{a}) \geq 1$ independent of $Q$ , $ε$ and $Q_{0}$ .

2.1step 1.1F3F4F6algebra

The reverse-exponent iteration. Let $κ = n / (n - 2)$ for $n \geq 3$ and fix $κ = 2$ for $n = 2$ , so the $L^{2 κ}$ Sobolev inequality is available in both cases by [F3]. Combining step 1.1 with the product rule and a cutoff equal to one on $B_{r}$ and supported in $B_{s}$ , $0 < r < s \leq 1$ , yields $∥ U^{- 1} ∥_{L^{p κ} (B_{r})} \leq (\frac{C}{s - r})^{2 / p} ∥ U^{- 1} ∥_{L^{p} (B_{s})}, C = C (n, θ, M_{a}) .$ For $p_{j} = p κ^{j}$ and $r_{j} = ρ + (1 - ρ) 2^{- j}$ , apply this with $(p_{j}, r_{j + 1}, r_{j})$ and multiply. The logarithm of the product is bounded by a constant multiple of $\sum_{j \geq 0} (1 + j) κ^{- j} < \infty$ , so ${ess sup}_{B_{ρ}} U^{- 1} \leq C_{1} ∥ U^{- 1} ∥_{L^{p} (B_{1})}, C_{1} = C_{1} (n, θ, M_{a}, p, ρ) .$ Taking reciprocals and inserting the volume factor gives $(\frac{1}{∣ B_{1} ∣} \int_{B_{1}} U^{- p} d x)^{- 1 / p} \leq C_{2} {ess inf}_{B_{ρ}} U$ . This is a positive-exponent iteration for $U^{- 1}$ ; in particular the reverse-exponent range is $p_{j} = p κ^{j} > 0$ , with arbitrary starting $p > 0$ .

2.2step 1.1step 1.2F1F4F6F7F8algebra

Bounded truncations, stopping cubes and factorial moments. For each $M > 0$ set $X_{M} : = \min {∣ w - ℓ ∣, M}$ . It is a bounded $H^{1}$ truncation by the Lipschitz chain rule; its mean oscillation on every dyadic subcube of a covering cube $Q_{0}$ is at most $2 K$ . By [F8], dyadic averages differentiate $X_{M}$ almost everywhere, so the stopping cubes cover the relevant superlevel set up to a null set. For $g = X_{M}$ and every dyadic cube $Q$ , ${avg}_{Q} ∣ g - g_{Q} ∣ \leq 2 K$ : compare first with the constant $\min {∣ w_{Q} - ℓ ∣, M}$ using the $1$ -Lipschitz scalar map, then with $g_{Q}$ . Set $A = 2^{n + 3} K$ . In each cube $Q$ select the maximal proper dyadic subcubes $P$ with ${avg}_{P} ∣ g - g_{Q} ∣ > A$ . They are disjoint and their total measure is at most $q ∣ Q ∣$ , where $q = 2 K / A = 2^{- (n + 2)}$ . Their immediate parents are not bad, so $∣ g_{P} - g_{Q} ∣ \leq {avg}_{P} ∣ g - g_{Q} ∣ \leq 2^{n} A$ . Outside their union, dyadic differentiation gives $∣ g - g_{Q} ∣ \leq A$ a.e. Repeat the same selection inside each selected cube, recentering at its own mean; its mean oscillation is still at most $2 K$ . The generation- $m$ union has measure at most $q^{m} ∣ Q_{0} ∣$ , while outside it the accumulated mean differences and final good-set bound give $∣ g - g_{Q_{0}} ∣ \leq m 2^{n} A$ for $m \geq 1$ . Consequently there are dimensional constants $C_{n}, c_{n} > 0$ such that $\frac{∣ {x \in Q_{0} : ∣ X_{M} - (X_{M})_{Q_{0}} ∣ > t} ∣}{∣ Q_{0} ∣} \leq C_{n} e^{- c_{n} t / K} .$ The layer-cake formula [F8] then yields $\frac{1}{∣ Q_{0} ∣} \int_{Q_{0}} ∣ X_{M} - (X_{M})_{Q_{0}} ∣^{j} \leq C_{n} j! (K / c_{n})^{j}$ for every integer $j \geq 1$ . The factorial cancels the denominator in the exponential series: its $j$ th averaged term is at most $C_{n} (p_{0} K / c_{n})^{j}$ . Choose $p_{0} : = \min {1, c_{n} / (2 K)} > 0$ . The geometric bound and monotone convergence of the nonnegative series give $\frac{1}{∣ Q_{0} ∣} \int_{Q_{0}} e^{p_{0} ∣ X_{M} - (X_{M})_{Q_{0}} ∣} d x \leq C_{n}^{'} .$ By step 1.2 and the fixed cube size, $(X_{M})_{Q_{0}} \leq \frac{1}{∣ Q_{0} ∣} \int_{Q_{0}} ∣ w - ℓ ∣ \leq C$ uniformly in $M$ , hence $\frac{1}{∣ Q_{0} ∣} \int_{Q_{0}} e^{p_{0} X_{M}} d x \leq e^{p_{0} C} C_{n}^{'}$ . Since $X_{M} ↑ ∣ w - ℓ ∣$ , monotone convergence gives $\frac{1}{∣ Q_{0} ∣} \int_{Q_{0}} e^{p_{0} ∣ w - ℓ ∣} d x \leq C_{3}$ . Summing over the finite cover and normalizing yields $\frac{1}{∣ B_{3 / 4} ∣} \int_{B_{3 / 4}} e^{p_{0} ∣ w - ℓ ∣} d x \leq C_{4}$ , uniformly in $ε$ . The bounded negative-power test in step 1.1 was placed in $H_{0}^{1}$ by compact-support smooth density; here the bounded logarithm truncations ensure every oscillation estimate is finite before the monotone limit.

3.1step 1.1step 2.1step 2.2F7algebra∎

The product constant and the limit $ε ↓ 0$ . Since $e^{\pm p_{0} (w - ℓ)} \leq e^{p_{0} ∣ w - ℓ ∣}$ , step 2.2 gives $(\frac{1}{∣ B_{3 / 4} ∣} \int_{B_{3 / 4}} U^{p_{0}} d x) (\frac{1}{∣ B_{3 / 4} ∣} \int_{B_{3 / 4}} U^{- p_{0}} d x) \leq C_{4}^{2},$ hence the second assertion with comparison constant $C_{4}^{2 / p_{0}}$ . For the fixed exponent $p_{0}$ , $U^{- p_{0}} ↑ u^{- p_{0}}$ and $U^{p_{0}} ↓ u^{p_{0}}$ as $ε ↓ 0$ ; dominated convergence for the positive moment (using $(u + 1)^{p_{0}} \leq u + 1$ since $p_{0} \leq 1$ on this bounded ball) and monotone convergence for the negative moment pass the product bound. Since $u > 0$ a.e., the limiting positive moment is strictly positive, so the product bound also shows that this particular negative moment is finite. For an arbitrary exponent $p > 0$ in the first assertion, monotone convergence passes $avg U^{- p}$ with its extended value; interpret $(+ \infty)^{- 1 / p} = 0$ , so the reciprocal negative-moment inequality remains valid without asserting finiteness. The same scaling as in step 1.1 restores arbitrary $R, x_{0}$ ; all constants depend only on the listed parameters, and no positive lower bound for $u$ is assumed.

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Weak Harnack inequality for nonnegative supersolutions

Statement

Assume Countable Choice and the Axiom of Choice. Let $n \geq 2$ , let $Ω \subseteq R^{n}$ be open, let $A$ and $L_{0}$ be as in De Giorgi local boundedness of homogeneous subsolutions, and let $F \in L_{l o c}^{q} (Ω)$ with $q > n / 2$ . Let $u \in H^{1} (Ω; R)$ satisfy $u \geq 0$ a.e. and $a_{0} (u, φ) \geq - \int_{Ω} F φ d x for every nonnegative φ \in C_{c}^{\infty} (Ω; R),$ i.e. $u$ is a nonnegative weak supersolution of $L_{0} u = - F$ . Then for every ball $B_{R} (x_{0})$ with $B_{2 R} (x_{0}) ⋐ Ω$ and every $0 < p < n / (n - 2)$ , $R^{- n / p} ∥ u ∥_{L^{p} (B_{R} (x_{0}))} \leq C ({ess inf}_{B_{R / 2} (x_{0})} u + R^{2 - n / q} ∥ F ∥_{L^{q} (B_{2 R} (x_{0}))}),$ with $C = C (n, q, θ, M_{a}, p)$ independent of $R$ and $x_{0}$ . For $n = 2$ every finite $p$ is allowed, with the critical Sobolev embedding in place of the $2^{*}$ embedding. The forcing term enters additively and cannot be dropped: the exponent range $0 < p < n / (n - 2)$ and the threshold $q > n / 2$ are the ones the iteration actually produces.

Facts & Assumptions

Given: Countable Choice and the Axiom of Choice; an open set $Ω \subseteq R^{n}$ , $n \geq 2$ ; measurable symmetric uniformly elliptic coefficients $A$ with constants $θ, M_{a}$ ; the principal operator $L_{0} u = - D_{i} (a^{i j} D_{j} u)$ with form $a_{0}$ ; a source $F \in L_{l o c}^{q} (Ω)$ , $q > n / 2$ ; a nonnegative $u \in H^{1} (Ω; R)$ with $a_{0} (u, φ) \geq - \int_{Ω} F φ d x$ for all nonnegative $φ \in C_{c}^{\infty} (Ω; R)$ ; a ball $B_{R} (x_{0})$ with $B_{2 R} (x_{0}) ⋐ Ω$ and $0 < p < n / (n - 2)$ when $n \geq 3$ , or any finite $p > 0$ when $n = 2$ .

[F1]

Assume the Axiom of Choice. Moser chains for positive supersolutions: if $w \in H^{1} (B_{S}; R)$ satisfies $w > 0$ a.e. and $a_{0} (w, v) \geq 0$ for every nonnegative $v \in H_{0}^{1} (B_{S})$ , then for every $0 < ρ < 1$ and every $p > 0$ , $(\frac{1}{∣ B_{S} ∣} \int_{B_{S}} w^{- p} d x)^{- 1 / p} \leq C_{1} {ess inf}_{B_{ρ S}} w$ , and for some $p_{0} = p_{0} (n, θ, M_{a}) > 0$ one has $(\frac{1}{∣ B_{3 S / 4} ∣} \int_{B_{3 S / 4}} w^{p_{0}} d x)^{1 / p_{0}} \leq C_{1} (\frac{1}{∣ B_{3 S / 4} ∣} \int_{B_{3 S / 4}} w^{- p_{0}} d x)^{- 1 / p_{0}}$ ; the constants depend only on their listed arguments (Moser iteration for positive supersolutions: negative-power and logarithmic comparison, The average of a locally integrable function over a Euclidean ball).

[F2]

Assume the Axiom of Choice. Logarithmic estimate: for every positive supersolution $w$ as in [F1] on a ball, every $η \in C_{c}^{\infty}$ and every $ε > 0$ , $\int η^{2} ∣ D \log (w + ε) ∣^{2} d x \leq \frac{4 M_{a}^{2}}{θ} \int ∣ D η ∣^{2} d x$ (Logarithmic Caccioppoli estimate for positive supersolutions).

[F3]

Assume the Axiom of Choice. On the reference ball $B_{2}$ , the weak maximum principle for a zero-trace solution of $L_{0} h = g$ gives $h \geq 0$ when $g \geq 0$ and ${ess sup}_{B_{2}} h \leq C ∥ g^{+} ∥_{L^{q} (B_{2})}$ for $q > n / 2$ (and $q > 1$ in dimension two). The constant is fixed for this ball, and the estimate applies after scaling $B_{2 R}$ to $B_{2}$ (Weak maximum principle for coercive divergence-form equations, The $L^{p}$ trace operator on a bounded $C^{1}$ domain, The kernel of the trace is the closure of the test functions, Bounded C^k domains and boundary charts).

[F4]

Assume the Axiom of Choice. Lax-Milgram and coercivity on $H_{0}^{1} (B_{2})$ : $H^{1} (B_{2})$ is Hilbert by $H^{k}$ is a Hilbert space under the derivative-sum inner product. The closure definition makes $H_{0}^{1} (B_{2})$ a closed linear subspace; a Cauchy sequence converges in $H^{1} (B_{2})$ and its limit remains in that closure, so the inherited inner product makes it Hilbert. The form $a_{0}$ is a bounded coercive form there, and every bounded conjugate-linear functional on $H_{0}^{1} (B_{2})$ is represented by a unique weak Dirichlet solution (The Lax--Milgram theorem, Coercivity of the principal Dirichlet form, The elliptic form is well defined and bounded on $H^{1}$ , The negative Sobolev space $H^{- 1} (Ω)$ , Weak Dirichlet solutions for a divergence-form operator, Zero-boundary Sobolev space as a norm closure, The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction).

[F5]

Assume the Axiom of Choice. Embedding and Holder input: for $n \geq 3$ and $1 \leq r < 2^{*}$ there is $C_{r}$ with $∥ v ∥_{L^{r} (B_{1})} \leq C_{r} ∥ v ∥_{H_{0}^{1} (B_{1})}$ ; in dimension two the same holds for every finite $r$ . By dilation this makes $φ \mapsto \int_{B_{2}} F^{+} φ$ bounded on $H_{0}^{1} (B_{2})$ when $q > n / 2$ (and $q > 1$ for $n = 2$ ), since the conjugate exponent $q^{'}$ lies in the available Sobolev range. Also, if $z \in H^{1} (B_{2})$ , multiplying by a smooth cutoff supported in $B_{2}$ and equal to one on $B_{3 / 2}$ gives $z \in L^{r} (B_{3 / 2})$ for every $1 \leq r < 2^{*}$ when $n \geq 3$ and every finite $r$ when $n = 2$ . The Sobolev norms scale as $∥ v ∥_{L^{r} (B_{R})} \leq C_{r} R^{1 + n / r - n / 2} ∥ D v ∥_{L^{2} (B_{R})}$ for $v \in H_{0}^{1} (B_{R})$ , with the corresponding inhomogeneous local estimate after cutoff; Holder's inequality gives $∥ g ∥_{L^{r} (E)} \leq ∣ E ∣^{1 / r - 1 / s} ∥ g ∥_{L^{s} (E)}$ for $1 \leq r < s \leq \infty$ (The Sobolev inequality for zero-boundary Sobolev closures on open sets, The critical Sobolev embedding into every finite $L^{q}$ , Holder's inequality for integrals, including the endpoint cases, The space $L^{p} (μ)$ as the quotient by null functions).

[F6]

Assume the Axiom of Choice. For $U = u + ε \geq ε > 0$ and $0 < s < 1$ , both scalar maps $t \mapsto (\max {t, 0} + ε)^{s - 1}$ and $t \mapsto (\max {t, 0} + ε)^{s / 2}$ are globally Lipschitz. Their compositions with $u$ lie in $H_{l o c}^{1}$ ; the first, multiplied by a compactly supported smooth cutoff squared, gives an $H_{0}^{1}$ test by the product rule, zero extension and smooth density. The second gives $V = U^{s / 2} \in H_{l o c}^{1}$ with $D V = (s / 2) U^{s / 2 - 1} D u$ (Chain rule for globally Lipschitz scalar maps of Sobolev functions, Weak Leibniz rule with a smooth factor, Compactly supported Sobolev functions extend by zero in every integer order, Compactly supported smooth functions are dense in W^{k,p}(R^n), Zero-boundary Sobolev space as a norm closure, Integer-order Sobolev spaces and their norms).

[F7]

Assume the Axiom of Choice. Dominated convergence passes integrals with an integrable majorant (Dominated convergence). Scaling invariance on doubled balls: with $v (y) : = u (x_{0} + R y)$ and $G (y) : = R^{2} F (x_{0} + R y)$ , the weak supersolution inequality scales to $B_{2}$ , $∥ G ∥_{L^{q} (B_{2})} = R^{2 - n / q} ∥ F ∥_{L^{q} (B_{2 R} (x_{0}))}$ , and $R^{- n / p} ∥ u ∥_{L^{p} (B_{R} (x_{0}))} = ∣ B_{1} ∣^{1 / p} (\frac{1}{∣ B_{1} ∣} \int_{B_{1}} ∣ v ∣^{p})^{1 / p}$ (Uniformly elliptic divergence-form operators and their sesquilinear forms, The average of a locally integrable function over a Euclidean ball, The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction).

Proof

technique · direct; scale the doubled ball to $B_2$, add a Lax-Milgram barrier there to make the solution a homogeneous supersolution on the full region required by Moser's comparison, derive the positive-integrability transitions using only negative-power tests with exponent $s-1<0$, then undo the scaling

1.1givenF3F4F5F7

Removing the source by a barrier on the doubled ball. After the rescaling of [F7] it suffices to treat $R = 1$ , $B_{2} (x_{0}) ⋐ Ω$ , and source norm $∥ F ∥_{L^{q} (B_{2})}$ . Since $F \in L^{q} (B_{2})$ and $q > n / 2$ , the local supersolution inequality extends by density from nonnegative smooth tests to all nonnegative $H_{0}^{1} (B_{2})$ tests: the embedding in [F5] puts $H_{0}^{1} (B_{2})$ in $L^{q^{'}} (B_{2})$ . The functional $φ \mapsto \int_{B_{2}} F^{+} φ d x$ is therefore bounded on $H_{0}^{1} (B_{2})$ , so [F4] gives a unique $h \in H_{0}^{1} (B_{2})$ with $a_{0} (h, φ) = \int_{B_{2}} F^{+} φ d x$ for all $φ \in H_{0}^{1} (B_{2})$ . Since $F^{+} \geq 0$ , [F3] gives $h \geq 0$ ; its radius-two estimate gives ${ess sup}_{B_{2}} h \leq C ∥ F^{+} ∥_{L^{q} (B_{2})} \leq C ∥ F ∥_{L^{q} (B_{2})}$ , where the fixed scaling factor $2^{2 - n / q}$ is absorbed into $C$ . With $w : = u + h$ one has $w \geq 0$ and $a_{0} (w, φ) \geq - \int_{B_{2}} F φ + \int_{B_{2}} F^{+} φ \geq 0$ for every nonnegative $φ \in H_{0}^{1} (B_{2})$ , so $w$ is a nonnegative homogeneous weak supersolution on the full doubled ball. Also $∥ u ∥_{L^{p} (B_{1})} \leq ∥ w ∥_{L^{p} (B_{1})}$ and ${ess inf}_{B_{1 / 2}} w \leq {ess inf}_{B_{1 / 2}} u + {ess sup}_{B_{2}} h$ .

1.2F1F2F5algebra

The seed exponent on a compactly contained ball. Let $w \geq 0$ be a homogeneous weak supersolution on $B_{2}$ and set $U : = w + ε$ . The outer ball $B_{7 / 4}$ is compactly contained in $B_{2}$ , so the comparison clause of [F1], supplied by the logarithmic estimate [F2], gives $(\frac{1}{∣ B_{21 / 16} ∣} \int_{B_{21 / 16}} U^{p_{0}})^{1 / p_{0}} \leq C_{1} (\frac{1}{∣ B_{21 / 16} ∣} \int_{B_{21 / 16}} U^{- p_{0}})^{- 1 / p_{0}}$ for some $p_{0} = p_{0} (n, θ, M_{a}) > 0$ . Apply the negative-power chain of [F1] with outer ball $B_{21 / 16} ⋐ B_{2}$ and ratio $ρ = 8 / 21$ ; it bounds the reciprocal negative moment on $B_{21 / 16}$ by $C_{2} {ess inf}_{B_{1 / 2}} U$ . Decrease the seed to $s_{0} : = \min {p_{0}, 1 / 2} < 1$ ; Jensen's inequality on the normalized ball mean gives $(\frac{1}{∣ B_{21 / 16} ∣} \int_{B_{21 / 16}} U^{s_{0}})^{1 / s_{0}} \leq C_{3} {ess inf}_{B_{1 / 2}} U$ .

1.3givenF3F5F6algebra

The positive-integrability transition for input exponents below one. Fix $0 < s < 1$ and a cutoff $η \in C_{c}^{\infty} (B_{R})$ with $η = 1$ on $B_{r}$ , $0 < r < R \leq 2$ . For $U = w + ε$ , the admissible test $η^{2} U^{s - 1}$ of [F6] and the homogeneous supersolution inequality give $(1 - s) \int η^{2} U^{s - 2} ⟨ A D w, D w ⟩ \leq 2 \int ∣ η ∣ U^{s - 1} ∣ ⟨ A D w, D η ⟩ ∣ .$ Cauchy--Schwarz in the $A$ -energy bounds the right side by $2 M_{a} (\int η^{2} U^{s - 2} ⟨ A D w, D w ⟩)^{1 / 2} (\int U^{s} ∣ D η ∣^{2})^{1 / 2}$ . Absorbing this energy square root and using ellipticity gives $\int η^{2} U^{s - 2} ∣ D w ∣^{2} \leq \frac{4 M_{a}^{2}}{(1 - s)^{2} θ} \int U^{s} ∣ D η ∣^{2}$ . With $V : = U^{s / 2}$ this becomes $\int η^{2} ∣ D V ∣^{2} \leq \frac{M_{a}^{2} s^{2}}{(1 - s)^{2} θ} \int V^{2} ∣ D η ∣^{2}$ . Applying Sobolev to $η V$ and the product rule therefore gives, for $n \geq 3$ and every $1 < λ \leq κ_{*} : = n / (n - 2)$ , or for $n = 2$ and every finite $λ > 1$ , $∥ U ∥_{L^{s λ} (B_{r})} \leq (\frac{C (s, λ, n, θ, M_{a})}{R - r})^{2 / s} ∥ U ∥_{L^{s} (B_{R})} .$ The truncation and density in [F6] justify the test; no estimate for an untruncated positive power is assumed.

2.1step 1.2step 1.3F5algebra

Reaching every exponent in the claimed range. Work on concentric balls between $B_{21 / 16}$ and $B_{1}$ , using equal positive radius gaps for the finitely many transitions below. If $0 < p \leq s_{0}$ , Jensen on $B_{1} \subset B_{21 / 16}$ and step 1.2 give $(\frac{1}{∣ B_{1} ∣} \int_{B_{1}} U^{p})^{1 / p} \leq C {ess inf}_{B_{1 / 2}} U$ . For $s_{0} < p \leq κ_{*} s_{0}$ when $n \geq 3$ , use step 1.3 once with $s = s_{0}$ and $λ = p / s_{0} \leq κ_{*}$ . If $κ_{*} s_{0} < p < κ_{*}$ , choose an integer $m \geq 1$ so large that $a : = (p / (κ_{*} s_{0}))^{1 / m} < κ_{*}$ . Apply step 1.3 $m$ times with exponent multiplier $a$ , reaching input exponent $s_{m} = s_{0} a^{m} = p / κ_{*} < 1$ , then once with multiplier $κ_{*}$ . Every input exponent is below one, so all tests in step 1.3 are admissible. The constants are finite and depend only on $n, θ, M_{a}, p$ . In dimension two, for any finite $p > s_{0}$ , a single use of step 1.3 with $s = s_{0}$ and finite $λ = p / s_{0}$ suffices. In every case this proves $(\frac{1}{∣ B_{1} ∣} \int_{B_{1}} U^{p})^{1 / p} \leq C {ess inf}_{B_{1 / 2}} U$ for the stated range.

3.1step 2.1F5F7

Removing regularization in the homogeneous case. Let $ε ↓ 0$ in step 2.1. The right side tends to $C {ess inf}_{B_{1 / 2}} w$ , while $U^{p} ↓ w^{p}$ and is dominated by $(w + 1)^{p}$ , integrable on $B_{3 / 2}$ by the cutoff-local Sobolev consequence in [F5] because $p < n / (n - 2) < 2^{*}$ for $n \geq 3$ and $p$ is finite for $n = 2$ (for $p < 1$ , use $(w + 1)^{p} \leq 1 + w$ ). Dominated convergence passes the positive-power mean and yields the homogeneous weak Harnack estimate.

4.1step 1.1step 3.1F5F7algebra∎

Conclusion with the source and the radius rescaling. For the barrier supersolution $w = u + h$ of step 1.1, step 3.1 gives $∥ w ∥_{L^{p} (B_{1})} \leq C {ess inf}_{B_{1 / 2}} w$ . Since $u \leq w$ and ${ess inf}_{B_{1 / 2}} w \leq {ess inf}_{B_{1 / 2}} u + {ess sup}_{B_{2}} h$ , step 1.1 gives $∥ u ∥_{L^{p} (B_{1})} \leq C ({ess inf}_{B_{1 / 2}} u + ∥ F ∥_{L^{q} (B_{2})})$ . Scaling back by [F7] gives the estimate with additive term $R^{2 - n / q} ∥ F ∥_{L^{q} (B_{2 R})}$ ; the doubled-ball hypothesis supplies the full region used in the barrier and in steps 1.2--2.1. The $n = 2$ argument allows every finite $p$ , and all constants are independent of $R, x_{0}$ .

Remarks

Radius convention. The quantitative interior form of the weak Harnack inequality controls the mean over $B_{R}$ by the essential infimum over $B_{R / 2}$ and requires the supersolution inequality on the doubled ball $B_{2 R}$ , exactly as in Theorem 2 of [K1] and Theorem 2 of [K2]; the statement records this explicitly rather than silently enlarging the class of admissible balls.

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Weak-Harnack exponent range and its dimension-dependent upper endpoint

Statement

The weak Harnack inequality of Weak Harnack inequality for nonnegative supersolutions is asserted only for the sourced range $0 < p < n / (n - 2)$ when $n \geq 3$ (and for every finite $p$ when $n = 2$ ). The endpoint $n / (n - 2)$ depends only on dimension; the seed exponent $p_{0}$ and constants depend on the coefficients. Starting from $p_{0}$ , the higher exponents below that endpoint are obtained by the positive-integrability Sobolev transitions in the weak-Harnack proof, while Hölder interpolation supplies smaller exponents. No claim is made here that every positive exponent is admissible; in particular one must not restate the weak Harnack inequality with an arbitrary $p > 0$ , and the constant for the source $F \in L^{q}$ depends on $n, q, θ, M_{a}, p$ as recorded.

Sources

Krummel, DeGiorgi-Nash lecture notes, Theorem 2 (printed p. 1) states the weak Harnack inequality for $0 < p < n / (n - 2)$ when $n \geq 3$ , and its proof obtains higher exponents from the fixed seed by Sobolev transitions, with Hölder interpolation giving smaller exponents. Simon, Lectures on Partial Differential Equations, Lecture 17, Theorem 2 (printed pp. 199-210) states the same range. The remark records the exact range of the theorem of Weak Harnack inequality for nonnegative supersolutions and carries no proof obligation of its own.

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Harnack inequality for nonnegative weak solutions

Statement

Assume Countable Choice and the Axiom of Choice. Let $n \geq 2$ , let $Ω \subseteq R^{n}$ be open, let $A$ and $L_{0}$ be as in De Giorgi local boundedness of homogeneous subsolutions, and let $F \in L_{l o c}^{q} (Ω)$ with $q > n / 2$ . Let $u \in H^{1} (Ω; R)$ satisfy $u \geq 0$ a.e. and be a weak solution of $L_{0} u = - F$ , i.e. $a_{0} (u, φ) = - \int_{Ω} F φ d x for every real φ \in C_{c}^{\infty} (Ω) .$ Then for every ball $B_{R} (x_{0})$ with $B_{2 R} (x_{0}) ⋐ Ω$ , ${ess sup}_{B_{R / 2} (x_{0})} u \leq C ({ess inf}_{B_{R / 2} (x_{0})} u + R^{2 - n / q} ∥ F ∥_{L^{q} (B_{2 R} (x_{0}))}), C = C (n, q, θ, M_{a}),$ independent of $R$ and $x_{0}$ ; in the homogeneous case $F = 0$ this is ${ess sup}_{B_{R / 2}} u \leq C {ess inf}_{B_{R / 2}} u$ , the Harnack inequality. For $n = 2$ every finite $q > 1$ is allowed. The two essential extrema are taken over the same ball, so no regularity of $u$ is needed for the statement; the additive forcing term is essential and the estimate is not claimed without it.

Facts & Assumptions

Given: Countable Choice and the Axiom of Choice; an open set $Ω \subseteq R^{n}$ , $n \geq 2$ ; uniformly elliptic measurable symmetric coefficients $A$ with constants $θ, M_{a}$ ; the principal operator $L_{0} u = - D_{i} (a^{i j} D_{j} u)$ with form $a_{0}$ ; a source $F \in L_{l o c}^{q} (Ω)$ , $q > n / 2$ ; a nonnegative $u \in H^{1} (Ω; R)$ with $a_{0} (u, φ) = - \int_{Ω} F φ d x$ for every real $φ \in C_{c}^{\infty} (Ω)$ ; a ball $B_{R} (x_{0})$ with $B_{2 R} (x_{0}) ⋐ Ω$ .

[F1]

Both roles of a local solution: the identity $a_{0} (u, φ) = - \int_{Ω} F φ$ against real compactly supported smooth tests gives both the subsolution inequality and the supersolution inequality for the equation $L_{0} u = - F$ , with the appropriate inequality directions (Weak subsolutions and supersolutions of a divergence-form equation, Uniformly elliptic divergence-form operators and their sesquilinear forms).

[F2]

Assume the Axiom of Choice. Local boundedness with a scale-correct source: for every ball $B_{S} (y) ⋐ Ω$ , every $0 < ρ < 1$ and every $p > 0$ , ${ess sup}_{B_{ρ S} (y)} w \leq C_{1} [(\frac{1}{∣ B_{S} (y) ∣} \int_{B_{S} (y)} w^{p})^{1 / p} + S^{2 - n / q} ∥ f^{+} ∥_{L^{q} (B_{S} (y))}]$ for every nonnegative weak subsolution $w$ of $a_{0} (w, \cdot) \leq \int f \cdot d x$ with $f \in L_{l o c}^{q} (Ω)$ , where $C_{1} = C_{1} (n, q, θ, M_{a}, ρ, p)$ (De Giorgi local boundedness with a scale-correct forcing term).

[F3]

Assume the Axiom of Choice. Weak Harnack inequality: for every ball $B_{S} (y)$ with $B_{2 S} (y) ⋐ Ω$ and every $0 < p < n / (n - 2)$ when $n \geq 3$ , or every finite $p > 0$ when $n = 2$ , $S^{- n / p} ∥ w ∥_{L^{p} (B_{S} (y))} \leq C_{2} ({ess inf}_{B_{S / 2} (y)} w + S^{2 - n / q} ∥ G ∥_{L^{q} (B_{2 S} (y))})$ for every nonnegative weak supersolution $w$ of $L_{0} w = - G$ with $G \in L_{l o c}^{q} (Ω)$ , where $C_{2} = C_{2} (n, q, θ, M_{a}, p)$ ; the range contains $s_{0} : = \min {p_{0}, 1 / 2}$ , where $p_{0} = p_{0} (n, θ, M_{a}) > 0$ is produced by the Moser iteration (Weak Harnack inequality for nonnegative supersolutions, Moser iteration for positive supersolutions: negative-power and logarithmic comparison).

[F4]

Assume the Axiom of Choice. Averaging and the elementary comparison of the negative part of the source: $\frac{1}{∣ B_{S} (y) ∣} \int_{B_{S} (y)} w^{p} d x = S^{- n} ∥ w ∥_{L^{p} (B_{S} (y))}^{p} / ∣ B_{1} ∣$ , and $∥ (- F)^{+} ∥_{L^{q}} = ∥ F^{-} ∥_{L^{q}} \leq ∥ F ∥_{L^{q}}$ (The average of a locally integrable function over a Euclidean ball, The space $L^{p} (μ)$ as the quotient by null functions, The essential supremum of a measurable function with respect to a measure).

[F5]

Assume the Axiom of Choice. The $n = 2$ substitute: the critical embedding $W_{0}^{1, 2} ↪ L^{κ}$ for every finite $κ$ replaces the $2^{*}$ embedding in both quoted theorems (The Sobolev inequality for zero-boundary Sobolev closures on open sets, The critical Sobolev embedding into every finite $L^{q}$ ).

Proof

technique · direct; combine the forcing local-boundedness estimate for the subsolution role of $u$ with the weak Harnack inequality for its supersolution role, both on the same ball $B_R$, and compare the two source terms

1.1givenF1F4

The forcing source in the subsolution role. By [F1] the solution $u$ is a nonnegative weak subsolution with source $- F$ , whose positive part is $(- F)^{+} = F^{-}$ ; by [F4], $∥ F^{-} ∥_{L^{q} (B_{R} (x_{0}))} \leq ∥ F ∥_{L^{q} (B_{2 R} (x_{0}))}$ .

2.1step 1.1F2F3F4

Chaining local boundedness with the weak Harnack inequality. Fix $s_{0} : = \min {p_{0}, 1 / 2}$ from [F3], which is an admissible weak-Harnack exponent in both the $n \geq 3$ and $n = 2$ ranges, and apply local boundedness [F2] to $u$ on $B_{R} (x_{0})$ with $ρ = 1 / 2$ . Its source term satisfies $R^{2 - n / q} ∥ F^{-} ∥_{L^{q} (B_{R})} \leq R^{2 - n / q} ∥ F ∥_{L^{q} (B_{2 R})}$ by [F4]. Apply weak Harnack [F3] to the supersolution $u$ on the same ball with $p = s_{0}$ and $G = F$ ; after converting the normalized mean to the stated norm, $(\frac{1}{∣ B_{R} ∣} \int_{B_{R}} u^{s_{0}})^{1 / s_{0}} \leq C_{2}^{'} ({ess inf}_{B_{R / 2}} u + R^{2 - n / q} ∥ F ∥_{L^{q} (B_{2 R})})$ , where $C_{2}^{'} = C_{2} / ∣ B_{1} ∣^{1 / s_{0}}$ . Substituting this bound into the local estimate gives coefficient $C_{1} C_{2}^{'}$ on the infimum and $C_{1} (C_{2}^{'} + 1)$ on the source term; thus $C : = C_{1} (C_{2}^{'} + 1)$ works for both and depends only on $n, q, θ, M_{a}$ .

3.1step 2.1F5algebra∎

The homogeneous case and the $n = 2$ clause. If $F = 0$ the same two steps give ${ess sup}_{B_{R / 2}} u \leq C_{1} C_{2}^{'} ({ess inf}_{B_{R / 2}} u)$ , the Harnack inequality; the extremal balls agree, so no regularity is used. For $n = 2$ the same proof applies with [F5] in place of the $2^{*}$ embedding in both quoted theorems and with every finite $p$ , so the range $0 < p < n / (n - 2)$ becomes unbounded. All arguments use Countable Choice and the Axiom of Choice only through the suppliers named above.

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A finite interior ball chain propagates weak Harnack bounds

Statement

Assume Countable Choice and the Axiom of Choice. Let $n \geq 3$ , let $Ω \subseteq R^{n}$ be open and connected, let $A, L_{0}$ be as in De Giorgi local boundedness of homogeneous subsolutions, let $F \in L_{l o c}^{q} (Ω)$ with $q > n / 2$ , and let $u \in H^{1} (Ω; R)$ with $u \geq 0$ a.e. be a weak solution of $L_{0} u = - F$ (Harnack inequality for nonnegative weak solutions). Let $K ⋐ Ω$ be compact and connected with positive Lebesgue measure. Then there are a number $N = N (K, Ω)$ and balls $B_{R_{1}} (x_{1}), \dots, B_{R_{N}} (x_{N}) ⋐ Ω$ with $⋃_{j = 1}^{N} B_{R_{j} / 2} (x_{j}) \supseteq K$ together with a constant $C = C (n, q, θ, M_{a}, K, Ω)$ such that ${ess sup}_{K} u \leq C ({ess inf}_{K} u + \sum_{j = 1}^{N} R_{j}^{2 - n / q} ∥ F ∥_{L^{q} (B_{2 R_{j}} (x_{j}))}) .$ The connectedness of $K$ makes the finite cover's overlap graph connected, and the constant grows with $N$ ; the forcing sum is finite because the cover is finite.

Facts & Assumptions

Given: Countable Choice and the Axiom of Choice; a connected open set $Ω \subseteq R^{n}$ ; uniformly elliptic measurable symmetric coefficients $A$ with constants $θ, M_{a}$ ; a source $F \in L_{l o c}^{q} (Ω)$ , $q > n / 2$ ; a nonnegative weak solution $u \in H^{1} (Ω; R)$ of $L_{0} u = - F$ ; a compact connected set $K ⋐ Ω$ with positive Lebesgue measure.

[F1]

Assume the Axiom of Choice. Harnack inequality on doubled balls: for every ball $B_{R} (x)$ with $B_{2 R} (x) ⋐ Ω$ , ${ess sup}_{B_{R / 2} (x)} u \leq C_{1} ({ess inf}_{B_{R / 2} (x)} u + R^{2 - n / q} ∥ F ∥_{L^{q} (B_{2 R} (x))})$ with $C_{1} = C_{1} (n, q, θ, M_{a})$ (Harnack inequality for nonnegative weak solutions, Weak subsolutions and supersolutions of a divergence-form equation).

[F2]

Assume the Axiom of Choice. Compactness and containment: since $K$ is compact and $Ω$ is open, $dist (K, R^{n} ∖ Ω) > 0$ , so a finite family of balls $B_{2 R_{i}} (x_{i}) ⋐ Ω$ , $x_{i} \in K$ , can be chosen with the half-balls $B_{R_{i} / 2} (x_{i})$ covering $K$ (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[F3]

Since $K$ is connected, the finite cover by the relative open sets $K \cap B_{R_{i} / 2} (x_{i})$ has a connected intersection graph: otherwise the unions corresponding to two components of the graph would separate $K$ . If two such relative open sets intersect, the corresponding open balls intersect in a nonempty open set and hence in a set of positive Lebesgue measure (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets).

[F4]

Assume Countable Choice. The zero extension of $u \in L^{2} (Ω)$ lies in $L^{2} (R^{n}) \subset L_{l o c}^{1} (R^{n})$ ; applying the cited Lebesgue-point theorem and restricting to $Ω$ gives a full-measure Lebesgue set (Lebesgue points and the Lebesgue set of an $L_{l o c}^{1}$ class, Almost every point is a Lebesgue point of a locally integrable function). At a Lebesgue point $x$ in a ball $B$ , ${ess inf}_{B} u \leq u (x) \leq {ess sup}_{B} u$ : if either inequality failed, the averages of $∣ u (y) - u (x) ∣$ over sufficiently small balls centered at $x$ would stay bounded below by a positive constant.

[F5]

For two measurable balls $B_{i}, B_{j}$ with $∣ B_{i} \cap B_{j} ∣ > 0$ , ${ess inf}_{B_{i}} u \leq {ess sup}_{B_{j}} u$ ; otherwise a real number strictly between them would be both an almost-everywhere lower bound on $B_{i}$ and an almost-everywhere upper bound on $B_{j}$ , impossible on their positive-measure intersection (The essential supremum of a measurable function with respect to a measure).

[F6]

If $M_{j} \leq C_{1} (M_{j + 1} + g_{j})$ for $j = 1, \dots, N - 1$ and $C_{1} \geq 1$ , then $M_{1} \leq C_{1}^{N} (M_{N} + \sum_{j = 1}^{N - 1} g_{j})$ by expanding the finite recurrence. [algebra]

Proof

technique · direct; cover $K$ by finitely many doubled balls whose half-balls have a connected overlap graph, propagate Harnack along a graph path, and use Lebesgue-point values at the endpoints to compare the essential extrema on $K$

1.1givenF2F3

The finite cover and connected overlap graph. By [F2] choose finitely many balls $B_{R_{i}} (x_{i})$ , $x_{i} \in K$ , with $B_{2 R_{i}} (x_{i}) ⋐ Ω$ whose half-balls cover $K$ . By [F3] their intersection graph is connected. Let $G : = \sum_{i = 1}^{N} g_{i}$ , where $g_{i} : = R_{i}^{2 - n / q} ∥ F ∥_{L^{q} (B_{2 R_{i}} (x_{i}))}$ ; this sum is finite because the cover is finite and $F \in L_{l o c}^{q} (Ω)$ .

2.1step 1.1F1F3F4F5F6

Endpoint estimate along a graph path. Put $C_{0} : = \max {1, C_{1}}$ , where $C_{1}$ is the local Harnack constant in [F1]. Let $x, y \in K$ be Lebesgue points of $u$ , and choose cover half-balls $B_{R_{i} / 2} (x_{i})$ and $B_{R_{j} / 2} (x_{j})$ containing them. By [F3] there is a path $i = i_{0}, i_{1}, \dots, i_{ℓ} = j$ in the finite intersection graph, with $ℓ \leq N - 1$ . Write $M_{s} : = {ess sup}_{B_{R_{i_{s}} / 2} (x_{i_{s}})} u$ and $m_{s} : = {ess inf}_{B_{R_{i_{s}} / 2} (x_{i_{s}})} u$ . For $s < ℓ$ , the Harnack bound [F1] and the correctly oriented overlap comparison [F5] give $M_{s} \leq C_{0} (m_{s} + g_{i_{s}}) \leq C_{0} (M_{s + 1} + g_{i_{s}})$ . Iterating by [F6] and applying [F1] on the last ball gives $u (x) \leq M_{0} \leq C_{0}^{ℓ + 1} (u (y) + \sum_{s = 0}^{ℓ} g_{i_{s}}) \leq C_{0}^{N} (u (y) + G)$ , because [F4] gives $u (x) \leq M_{0}$ and $m_{ℓ} \leq u (y)$ at Lebesgue points.

3.1step 2.1F4algebra∎

Conclusion for essential extrema on $K$ . The set of Lebesgue points in $K$ has full measure in $K$ by [F4]. For any $ε > 0$ , the positive-measure hypothesis on $K$ and the definition of essential infimum give a Lebesgue point $y \in K$ with $u (y) < {ess inf}_{K} u + ε$ . Applying step 2.1 with this fixed $y$ gives $u (x) \leq C_{0}^{N} ({ess inf}_{K} u + ε + G)$ for almost every Lebesgue point $x \in K$ . Taking the essential supremum over $K$ and then letting $ε ↓ 0$ proves the stated inequality with $C : = C_{0}^{N}$ .

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Zero-set propagation for a nonnegative Holder weak solution

Statement

Assume Countable Choice and the Axiom of Choice. Let $n \geq 3$ , let $Ω \subseteq R^{n}$ be connected and open, let $A, L_{0}$ be as in De Giorgi local boundedness of homogeneous subsolutions, and let $u \in H^{1} (Ω; R)$ with $u \geq 0$ a.e. be a weak solution of $L_{0} u = 0$ . Let $u^{*}$ be a continuous representative of $u$ on $Ω$ (such a representative exists by De Giorgi-Nash interior Holder regularity for divergence-form equations) and let $x_{0} \in Ω$ with $u^{*} (x_{0}) = 0$ . Then $u^{*} \equiv 0$ on $Ω$ ; equivalently the zero set ${u^{*} = 0}$ is both relatively open and relatively closed in $Ω$ . The same argument shows: if $u \in H^{1} (Ω)$ is merely a nonnegative weak supersolution of $L_{0} u \geq 0$ and a representative of $u$ is continuous at a point $x_{0}$ with value $0$ , then $u = 0$ a.e. on a neighbourhood of $x_{0}$ ; the global conclusion then needs a continuous representative on all of $Ω$ .

Facts & Assumptions

Given: Countable Choice and the Axiom of Choice; a connected open set $Ω \subseteq R^{n}$ , $n \geq 3$ ; uniformly elliptic measurable symmetric coefficients $A$ with constants $θ, M_{a}$ ; the principal operator $L_{0} u = - D_{i} (a^{i j} D_{j} u)$ with form $a_{0}$ ; a nonnegative weak solution $u \in H^{1} (Ω; R)$ of $L_{0} u = 0$ ; a continuous representative $u^{*}$ of $u$ ; a point $x_{0} \in Ω$ with $u^{*} (x_{0}) = 0$ .

[F1]

Assume the Axiom of Choice. Weak Harnack inequality at $p = 1$ : for every ball $B_{S} (y)$ with $B_{2 S} (y) ⋐ Ω$ one has $S^{- n} ∥ w ∥_{L^{1} (B_{S} (y))} \leq C_{1} ({ess inf}_{B_{S / 2} (y)} w + S^{2 - n / q} ∥ F ∥_{L^{q} (B_{2 S} (y))})$ for every nonnegative supersolution $w$ of $L_{0} w = - F$ with $F \in L_{l o c}^{q}$ , $q > n / 2$ , with $C_{1} = C_{1} (n, q, θ, M_{a})$ (Weak Harnack inequality for nonnegative supersolutions).

[F2]

Assume the Axiom of Choice. A class in $H^{1}$ with $\int_{B} ∣ w ∣ d x = 0$ on an open ball $B$ vanishes a.e. on $B$ ; and the essential infimum of a nonnegative class over a ball is the infimum of any continuous representative over that ball, so that ${ess inf}_{B_{S / 2} (z)} u \leq u^{*} (x_{0}) = 0$ whenever $x_{0} \in B_{S / 2} (z)$ and $u \geq 0$ a.e. (The space $L^{p} (μ)$ as the quotient by null functions, The average of a locally integrable function over a Euclidean ball, The essential supremum of a measurable function with respect to a measure, Local Hölder and scaled C-two-alpha norms on balls).

[F3]

Assume the Axiom of Choice. Continuity and connectedness: the zero set of a continuous function is relatively closed, and a nonempty subset of a connected topological space that is both relatively open and relatively closed is the whole space (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets, Local Hölder and scaled C-two-alpha norms on balls).

[F4]

Assume the Axiom of Choice. Supersolution and subsolution vocabulary: a weak solution of $L_{0} u = 0$ is in particular a nonnegative weak supersolution of $L_{0} u \geq 0$ , and $L_{0} u = 0$ weakly means $a_{0} (u, v) = 0$ for every $v \in H_{0}^{1} (Ω)$ (Weak subsolutions and supersolutions of a divergence-form equation, Uniformly elliptic divergence-form operators and their sesquilinear forms).

Proof

technique · direct; the weak Harnack inequality with $p=1$ turns the vanishing of $u^*$ at a point into the vanishing of $u$ on a whole ball, which makes the zero set relatively open, while continuity makes it relatively closed; connectedness then forces the zero set to be all of $\Omega$

1.1givenF1F2

The zero set is relatively open. Fix a radius $S > 0$ with $B_{2 S} (x_{0}) ⋐ Ω$ ; such an $S$ exists because $Ω$ is open and $x_{0} \in Ω$ . Apply the weak Harnack inequality [F1] to $u$ on the ball $B_{S} (x_{0})$ with $p = 1$ and $F = 0$ : $S^{- n} \int_{B_{S} (x_{0})} u d x \leq C_{1} {ess inf}_{B_{S / 2} (x_{0})} u$ . Since $x_{0} \in B_{S / 2} (x_{0})$ and $u \geq 0$ a.e. with continuous representative $u^{*}$ vanishing at $x_{0}$ , [F2] gives ${ess inf}_{B_{S / 2} (x_{0})} u \leq u^{*} (x_{0}) = 0$ ; hence $\int_{B_{S} (x_{0})} u d x = 0$ and therefore $u = 0$ a.e. on $B_{S} (x_{0})$ by [F2]. Since $u^{*}$ is continuous and agrees with $u$ a.e. on the ball $B_{S} (x_{0})$ , the set where $u^{*} \neq 0$ is open and of measure zero in $B_{S} (x_{0})$ ; it must be empty, so $u^{*} = 0$ on all of $B_{S} (x_{0})$ .

2.1step 1.1F1F2F3F4

The zero set is relatively closed and the second assertion. The set $Z : = {x \in Ω : u^{*} (x) = 0}$ is the preimage of the closed set ${0}$ under the continuous map $u^{*}$ , hence relatively closed in $Ω$ by [F3]. For the second assertion, suppose only that $u \in H^{1} (Ω)$ is a nonnegative weak supersolution of $L_{0} u \geq 0$ and that a representative is continuous at $x_{0}$ with value $0$ ; then the same computation with $F = 0$ and the continuity of the representative at the single point $x_{0}$ gives $\int_{B_{S} (x_{0})} u d x = 0$ for some $S > 0$ , hence $u = 0$ a.e. on $B_{S} (x_{0})$ , which is the local conclusion; the global conclusion needs a representative continuous on all of $Ω$ so that [F3] applies to the whole zero set.

3.1step 2.1F3∎

Conclusion by connectedness. The set $Z$ is nonempty (it contains $x_{0}$ ), relatively open by step 1.1 and relatively closed by step 2.1; since $Ω$ is connected, [F3] gives $Z = Ω$ , that is $u^{*} \equiv 0$ on $Ω$ , which is the assertion. All arguments use Countable Choice and the Axiom of Choice only through the suppliers named above.

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Strong maximum principle for weak elliptic solutions

Statement

Assume Countable Choice and the Axiom of Choice. Let $n \geq 3$ , let $Ω \subseteq R^{n}$ be connected and open, let $A, L_{0}$ be as in De Giorgi local boundedness of homogeneous subsolutions, and let $u \in H^{1} (Ω; R)$ with $u \geq 0$ a.e. be a weak solution of $L_{0} u = 0$ on $Ω$ . Then either $u = 0$ a.e. on $Ω$ , or $u > 0$ a.e. on $Ω$ ; moreover the Holder representative $u^{*}$ of De Giorgi-Nash interior Holder regularity for divergence-form equations satisfies: if $u^{*}$ vanishes at one point of $Ω$ , then $u^{*} \equiv 0$ on $Ω$ , and otherwise $u^{*} > 0$ on all of $Ω$ . In particular a nonnegative weak solution that is not identically zero is strictly positive after the representative is fixed, and no interior zero is possible.

Facts & Assumptions

Given: Countable Choice and the Axiom of Choice; a connected open set $Ω \subseteq R^{n}$ , $n \geq 3$ ; uniformly elliptic measurable symmetric coefficients $A$ with constants $θ, M_{a}$ ; the principal operator $L_{0} u = - D_{i} (a^{i j} D_{j} u)$ ; a nonnegative weak solution $u \in H^{1} (Ω; R)$ of $L_{0} u = 0$ ; a Holder representative $u^{*}$ of $u$ on $Ω$ .

[F1]

Assume the Axiom of Choice. Zero-set propagation: if $u \geq 0$ a.e. is a weak solution of $L_{0} u = 0$ on a connected open set and $u^{*}$ is a continuous representative with $u^{*} (x_{0}) = 0$ for some $x_{0} \in Ω$ , then $u^{*} \equiv 0$ on $Ω$ (Zero-set propagation for a nonnegative Holder weak solution, Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets).

[F2]

Assume the Axiom of Choice. The representative exists, is continuous, and agrees with $u$ almost everywhere, so $u^{*} \geq 0$ everywhere because $u \geq 0$ a.e. and $u^{*}$ is continuous; conversely if $u^{*} > 0$ on $Ω$ then $u > 0$ a.e. (De Giorgi-Nash interior Holder regularity for divergence-form equations, Local Hölder and scaled C-two-alpha norms on balls, The space $L^{p} (μ)$ as the quotient by null functions).

[F3]

Assume the Axiom of Choice. Weak solution vocabulary: $L_{0} u = 0$ weakly means $a_{0} (u, v) = 0$ for every $v \in H_{0}^{1} (Ω)$ , and in particular $u$ is both a weak subsolution and a weak supersolution (Weak subsolutions and supersolutions of a divergence-form equation, Uniformly elliptic divergence-form operators and their sesquilinear forms).

Proof

technique · direct dichotomy; either the continuous representative vanishes somewhere, in which case the zero-set propagation lemma makes it identically zero, or it has no zero, in which case continuity and nonnegativity make it strictly positive

1.1givenF1F2

The case of a zero. If there is $x_{0} \in Ω$ with $u^{*} (x_{0}) = 0$ , then [F1] gives $u^{*} \equiv 0$ on the connected set $Ω$ ; since $u = u^{*}$ a.e., $u = 0$ a.e. on $Ω$ .

2.1step 1.1F2F3∎

The case of no zero. If $u^{*}$ vanishes nowhere on $Ω$ , then $u^{*} \neq 0$ everywhere; by [F2] $u^{*} \geq 0$ everywhere, so $u^{*} > 0$ on all of $Ω$ , and hence $u > 0$ a.e. on $Ω$ . Thus either $u = 0$ a.e. or $u > 0$ a.e.; in the first case $u^{*} \equiv 0$ (as the continuous representative of the zero class) and in the second $u^{*} > 0$ everywhere. In particular no point of $Ω$ can be an interior zero of $u^{*}$ unless $u^{*}$ vanishes identically. All arguments use Countable Choice and the Axiom of Choice only through the suppliers named above.

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Scalar De Giorgi theory does not transfer verbatim to systems

Statement

The De Giorgi--Nash--Moser estimates on this page concern a single real-valued unknown. In De Giorgi-Nash interior Holder regularity for divergence-form equations, $u \in H^{1} (Ω; R)$ solves the scalar equation $- div (A \nabla u) = 0$ , where $A$ is a measurable symmetric uniformly elliptic spatial matrix. These are the hypotheses in [V] §1, Theorem 1. A component of a coupled elliptic system need not satisfy this scalar equation, so the theorem cannot be applied to that component merely because the system has an ellipticity condition. In particular, a system condition such as Legendre--Hadamard ellipticity does not by itself check the scalar hypotheses of this page. Any application to components must separately verify those hypotheses, as one can for a decoupled collection of scalar equations. Regularity theory for coupled systems is outside this page's scope.

Sources

Velichkov, Elliptic PDEs: Teorema di De Giorgi, Section 1 and Theorem 1 (printed p. 1 of the complete 7-page note, read in full), states and proves the interior regularity theorem for a scalar real-valued solution $u$ of $- div (A \nabla u) = 0$ with a symmetric uniformly elliptic matrix $A$ ; the statement has no vector-valued or system analogue. This item records only the resulting limitation of De Giorgi-Nash interior Holder regularity for divergence-form equations and asserts no system counterexample and no system regularity theorem.

5 · Examples, counterexamples and false statements

None yet.