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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.

Depends on

Used by

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Sources

Brian Krummel, DeGiorgi-Nash lecture notes (15 March 2016; complete 9-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)