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The Caccioppoli inequality for weak elliptic solutions

Statement

Assume Countable Choice. Let $Ω \subseteq R^{n}$ be open, $n \geq 1$ , $K \in {R, C}$ , let $L, a$ be as in Uniformly elliptic divergence-form operators and their sesquilinear forms with ellipticity constant $θ$ and coefficient bounds $M_{a}, M_{b}, M_{c}$ , let $f \in L_{l o c}^{2} (Ω)$ and let $u \in H^{1} (Ω; K)$ be a local weak solution of $L u = f$ on $Ω$ (Local weak solutions of a divergence-form operator). For every ball $B_{R} (x_{0}) ⋐ Ω$ and every $0 < r < R$ there is a constant $C = C (n, θ, M_{a}, M_{b}, M_{c})$ with $\int_{B_{r} (x_{0})} ∣ D u ∣^{2} d x \leq C (\frac{1}{(R - r)^{2}} \int_{B_{R} (x_{0})} ∣ u ∣^{2} d x + \int_{B_{R} (x_{0})} ∣ u ∣^{2} d x + \int_{B_{R} (x_{0})} ∣ f ∣^{2} d x) .$ No regularity of the coefficients beyond measurability and essential boundedness is used, and the estimate is uniform in the localisation. When $f = 0$ the estimate is the energy inequality for a locally weak harmonic class.

Facts & Assumptions

Given: Countable Choice; an open set $Ω \subseteq R^{n}$ with $n \geq 1$ ; a scalar field $K \in {R, C}$ ; coefficients $a^{i j}, b^{i}, c$ and the form $a (\cdot, \cdot)$ of Uniformly elliptic divergence-form operators and their sesquilinear forms with ellipticity constant $θ$ and bounds $M_{a}, M_{b}, M_{c}$ ; a class $f \in L_{l o c}^{2} (Ω)$ ; a local weak solution $u \in H^{1} (Ω)$ of $L u = f$ ; a ball $B_{R} (x_{0}) ⋐ Ω$ ; and a radius $0 < r < R$ ; the standard smooth step $σ$ of The standard smooth step function is fixed once and for all, with $S : = ∥ σ^{'} ∥_{L^{\infty} (R)}$ .

[F1]

The form is well defined on $H^{1} (Ω)$ and bounded: $∣ a (w, z) ∣ \leq (n M_{a} + n M_{b} + M_{c}) ∥ w ∥_{H^{1}} ∥ z ∥_{H^{1}}$ for $w, z \in H^{1} (Ω)$ , the value depends only on the $H^{1}$ classes, and the form is linear in the first and conjugate-linear in the second slot. (The elliptic form is well defined and bounded on $H^{1}$ , Uniformly elliptic divergence-form operators and their sesquilinear forms)

[F2]

Uniform ellipticity: $Re (\sum_{i, j} a^{i j} (x) ξ_{j} \overline{ξ_{i}}) \geq θ ∣ ξ ∣^{2}$ for almost every $x \in Ω$ and every $ξ \in C^{n}$ ; the coefficient bounds $∣ a^{i j} ∣ \leq M_{a}$ , $∣ b^{i} ∣ \leq M_{b}$ , $∣ c ∣ \leq M_{c}$ hold almost everywhere. (Uniformly elliptic divergence-form operators and their sesquilinear forms)

[F3]

The local weak equation is equivalent to $a (u, v) = \int_{Ω_{2}} f \overline{v} d x$ for every bounded open $Ω_{2} ⋐ Ω$ and every $v \in H_{0}^{1} (Ω_{2})$ , and every class in $H_{0}^{1} (Ω_{2})$ may be used as a test class there. (Local weak solutions of a divergence-form operator, Zero-boundary Sobolev space as a norm closure)

[F4]

Smooth-factor Leibniz rule: for $η \in C_{c}^{\infty} (Ω)$ and $u \in H^{1} (Ω)$ the class $η^{2} u$ lies in $H^{1} (Ω)$ and $D_{i} (η^{2} u) = η^{2} D_{i} u + 2 η (D_{i} η) u$ almost everywhere; a class in $H^{1} (Ω)$ with support in a compact subset of $Ω$ lies in $H_{0}^{1}$ of any open set containing its support. (Weak Leibniz rule with a smooth factor, Compactly supported Sobolev functions extend by zero in every integer order, Zero-boundary Sobolev space as a norm closure, The cutoff difference-quotient commutator estimate).

[F5]

The standard smooth step is smooth with values in $[0, 1]$ , vanishes on $(- \infty, 0]$ and equals $1$ on $[1, \infty)$ ; the chain rule computes the derivatives of $x \mapsto σ (g (x))$ as $σ^{'} (g (x)) D g (x)$ . (The standard smooth step function, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ )

[F6]

Cauchy-Schwarz and Young: for real vectors or scalars one has $∣ X Y ∣ \leq δ X^{2} + Y^{2} / (4 δ)$ for every $δ > 0$ , and $\int ∣ F G ∣ \leq ∥ F ∥_{L^{2}} ∥ G ∥_{L^{2}}$ ; the vector estimate $\sum_{i, j} ∣ a^{i j} D_{j} u \overline{D_{i} v} ∣ \leq n M_{a} ∣ D u ∣ ∣ D v ∣$ holds almost everywhere. (Young's inequality for conjugate real exponents, Holder's inequality for integrals, including the endpoint cases, Conjugate exponents, including the endpoint conventions)

Proof

technique · direct

1.1F5givenconstruct

Put $s : = (r + R) / 2$ and define $u_{*} (x) : = (s^{2} - ∣ x - x_{0} ∣^{2}) / (s^{2} - r^{2})$ and $η (x) : = σ (u_{*} (x))$ on $R^{n}$ . Then $η$ is smooth, $0 \leq η \leq 1$ , $η = 1$ on $B_{r} (x_{0})$ , and $supp η \subseteq \overline{B_{s} (x_{0})} \subset B_{R} (x_{0})$ : indeed $u_{*} \geq 1$ on $B_{r} (x_{0})$ and $u_{*} \leq 0$ off $B_{s} (x_{0})$ , while $s^{2} - r^{2} > 0$ . Thus the support is compactly contained in $B_{R} (x_{0}) ⋐ Ω$ . Moreover, on the support of $σ^{'} \circ u_{*}$ one has $0 \leq u_{*} \leq 1$ , hence $∣ x - x_{0} ∣ \leq s$ , and the chain rule gives $∣ D η (x) ∣ = ∣ σ^{'} (u_{*} (x)) ∣ ∣ D u_{*} (x) ∣ \leq S \frac{2 ∣ x - x_{0} ∣}{s^{2} - r^{2}} \leq S \frac{2 s}{(R - r) s / 2} = \frac{4 S}{R - r},$ because $s - r = (R - r) / 2$ and $s + r \geq s$ give $s^{2} - r^{2} = (s - r) (s + r) \geq (R - r) s / 2$ .

2.1F1F3F4step 1.1

The class $v : = η^{2} u$ lies in $H^{1} (Ω)$ by [F4] and its support is contained in $supp η \subseteq \overline{B_{s} (x_{0})} ⋐ B_{R} (x_{0}) ⋐ Ω$ , so $v \in H_{0}^{1} (B_{R} (x_{0}))$ by [F4]; since $B_{R} (x_{0}) ⋐ Ω$ is bounded, the weak equation of [F3] with the test class $v$ reads $a (u, η^{2} u) = \int_{B_{R} (x_{0})} f \overline{η^{2} u} d x$ , both sides finite by the boundedness of the form in [F1].

3.1F2F4F6step 2.1algebra

By the Leibniz rule of [F4], $D_{i} (η^{2} u) = η^{2} D_{i} u + 2 η (D_{i} η) u$ almost everywhere; substituting this into the definition of the form and splitting the principal part, $a (u, η^{2} u) = \int_{B_{R}} η^{2} a^{i j} D_{j} u \overline{D_{i} u} d x + \int_{B_{R}} 2 η a^{i j} D_{j} u \overline{(D_{i} η) u} d x + \int_{B_{R}} (b^{i} D_{i} u + c u) \overline{η^{2} u} d x,$ and taking real parts in the identity of step 2.1 gives $Re \int_{B_{R}} η^{2} a^{i j} D_{j} u \overline{D_{i} u} d x \leq 2 n M_{a} \int_{B_{R}} η ∣ D η ∣ ∣ D u ∣ ∣ u ∣ + n M_{b} \int_{B_{R}} η^{2} ∣ D u ∣ ∣ u ∣ + M_{c} \int_{B_{R}} η^{2} ∣ u ∣^{2} + \int_{B_{R}} η^{2} ∣ f ∣ ∣ u ∣,$ using [F2] for the left side ( $Re (a^{i j} D_{j} u \overline{D_{i} u}) \geq θ ∣ D u ∣^{2}$ ) and the coefficient bounds together with [F6] for each remaining term.

4.1F6step 3.1algebra

Estimate the four terms on the right of step 3.1 by Young's inequality with a parameter $δ > 0$ : $2 n M_{a} \int η ∣ D η ∣ ∣ D u ∣ ∣ u ∣ \leq 2 n M_{a} δ \int η^{2} ∣ D u ∣^{2} + \frac{2 n M_{a}}{4 δ} \int ∣ D η ∣^{2} ∣ u ∣^{2}$ ; $n M_{b} \int η^{2} ∣ D u ∣ ∣ u ∣ \leq n M_{b} δ \int η^{2} ∣ D u ∣^{2} + \frac{n M_{b}}{4 δ} \int η^{2} ∣ u ∣^{2}$ ; $\int η^{2} ∣ f ∣ ∣ u ∣ \leq δ \int η^{2} ∣ f ∣^{2} + \frac{1}{4 δ} \int η^{2} ∣ u ∣^{2}$ ; and $M_{c} \int η^{2} ∣ u ∣^{2}$ needs no splitting. Choosing $δ : = θ / (2 (2 n M_{a} + n M_{b}))$ makes the two $∣ D u ∣^{2}$ coefficients sum to at most $θ / 2$ , so the left side of step 3.1 controls the gradient: $\frac{θ}{2} \int_{B_{R}} η^{2} ∣ D u ∣^{2} d x \leq C_{1} \int_{B_{R}} ∣ D η ∣^{2} ∣ u ∣^{2} d x + C_{2} \int_{B_{R}} ∣ u ∣^{2} d x + C_{3} \int_{B_{R}} ∣ f ∣^{2} d x$ with constants $C_{1}, C_{2}, C_{3}$ depending only on $n, θ, M_{a}, M_{b}, M_{c}$ .

5.1step 1.1step 4.1algebra∎

Since $η = 1$ on $B_{r} (x_{0})$ and $supp η \subseteq B_{R} (x_{0})$ , one has $\int_{B_{r}} ∣ D u ∣^{2} \leq \int_{B_{R}} η^{2} ∣ D u ∣^{2}$ , and step 4.1 combined with the gradient bound $∣ D η ∣ \leq 4 S / (R - r)$ of step 1.1 gives $\int_{B_{r} (x_{0})} ∣ D u ∣^{2} d x \leq \frac{2}{θ} (C_{1} \frac{16 S^{2}}{(R - r)^{2}} \int_{B_{R} (x_{0})} ∣ u ∣^{2} d x + C_{2} \int_{B_{R} (x_{0})} ∣ u ∣^{2} d x + C_{3} \int_{B_{R} (x_{0})} ∣ f ∣^{2} d x),$ which is the displayed estimate with $C = \max {32 S^{2} C_{1} / θ, 2 C_{2} / θ, 2 C_{3} / θ} = C (n, θ, M_{a}, M_{b}, M_{c})$ , because $σ$ is fixed in advance and $S$ is a universal constant.

Source notes

Simon's Lecture 6, Lemma 1 (printed pp. 58-59) proves the estimate by testing with $η^{2} u$ ; Hunter's final step of Theorem 4.27 (printed pp. 112-114) uses the same test function. The explicit $(R - r)^{- 2}$ scale above comes from the rescaled standard bump, whose gradient bound is computed in step 1.1 rather than quoted as a separate lemma, and the constant is uniform over the choice of ball because no quantity depending on $x_{0}$ , $r$ or $R$ enters it except through the displayed powers.

Depends on

Used by

Scaled Caccioppoli inequality on concentric balls Corollary

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Sources

Leon Simon, Lectures on Partial Differential Equations (Stanford, complete 223-page author scan) (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (UC Davis, revised 18 June 2014, complete 242-page two-quarter graduate notes) (standard reference, not scraped)