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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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Sources

Bozhidar Velichkov, Elliptic PDEs: Teorema di De Giorgi (Universita di Pisa; complete 7-page note, in Italian) (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)
Brian Krummel, DeGiorgi-Nash lecture notes (15 March 2016; complete 9-page notes) (standard reference, not scraped)