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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.

Depends on

Used by

Strong maximum principle for weak elliptic solutions Corollary
Degenerate ellipticity allows nonconstant solutions with interior zero sets Counterexample

Dependency tree · two levels

82 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Brian Krummel, Consequences of De Giorgi-Nash-Moser (4 March 2016; complete 7-page notes) (standard reference, not scraped)
Leon Simon, Lectures on Partial Differential Equations (Stanford University; complete author scan, 118 sheets reproducing the 223 printed pages of the manuscript, two logical pages per sheet) (standard reference, not scraped)