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Interior $H^{2}$ estimate for constant-coefficient elliptic equations

Statement

Assume Countable Choice. Let $n \geq 1$ , $K \in {R, C}$ , let $a^{i j} \in K$ be a constant matrix satisfying $Re (\sum a^{i j} ξ_{j} \overline{ξ_{i}}) \geq θ ∣ ξ ∣^{2}$ with $θ > 0$ and $∣ a^{i j} ∣ \leq M_{a}$ , let $b^{i}, c \in K$ be constants with $∣ b^{i} ∣ \leq M_{b}$ , $∣ c ∣ \leq M_{c}$ , and let $L$ be the associated constant-coefficient divergence-form operator with form $a$ . Let $f \in L^{2} (B_{R} (x_{0}))$ and let $u \in H^{1} (B_{R} (x_{0}))$ solve $L u = f$ weakly on $B_{R} (x_{0})$ . Then $u \in H^{2} (B_{r} (x_{0}))$ for every $0 < r < R$ , with $∥ D^{2} u ∥_{L^{2} (B_{r} (x_{0}))} \leq C ((1 + \frac{1}{(R - r)^{2}}) ∥ u ∥_{L^{2} (B_{R} (x_{0}))} + ∥ f ∥_{L^{2} (B_{R} (x_{0}))}),$ where $C = C (n, θ, M_{a}, M_{b}, M_{c})$ . No symmetry of $a$ and no boundary condition on $u$ is required, and the estimate is the transparent constant-coefficient core of the variable-coefficient theorem.

Facts & Assumptions

Given: Countable Choice; the ball $B_{R} (x_{0})$ with $0 < r < R$ ; constant coefficients $a^{i j}, b^{i}, c$ with the bounds and ellipticity of the Statement; $f \in L^{2} (B_{R} (x_{0}))$ ; and a local weak solution $u \in H^{1} (B_{R} (x_{0}))$ of $L u = f$ .

[F1]

Local weak solution: $a (u, φ) = \int_{B_{R} (x_{0})} f \overline{φ} d x$ for every $φ \in C_{c}^{\infty} (B_{R} (x_{0}))$ . (Local weak solutions of a divergence-form operator)

[F2]

The constant-coefficient form and ellipticity: $a (w, z) = \int (a^{i j} D_{j} w \overline{D_{i} z} + b^{i} D_{i} w \overline{z} + c w \overline{z}) d x$ with $Re (a^{i j} ξ_{j} \overline{ξ_{i}}) \geq θ ∣ ξ ∣^{2}$ , $∣ b^{i} ∣ \leq M_{b}$ , $∣ c ∣ \leq M_{c}$ . (Uniformly elliptic divergence-form operators and their sesquilinear forms)

[F3]

Difference-quotient calculus: for locally integrable classes the two-domain integration-by-parts identity holds and reduces to $\int δ_{h}^{i} w \overline{φ} d x = - \int w \overline{δ_{- h}^{i} φ} d x$ whenever the product has compact support of distance exceeding $∣ h ∣$ from the boundary; the product rule $δ_{h}^{i} (w ζ) = (τ_{- h e_{i}} w) δ_{h}^{i} ζ + (δ_{h}^{i} w) ζ$ holds; and difference quotients commute with weak derivatives, $D^{α} (δ_{h}^{i} w) = δ_{h}^{i} (D^{α} w)$ on the shrunken domain. (Difference-quotient calculus: integration by parts, product rule, commutation)

[F4]

The localised test class: for $u \in H^{1}$ , $η \in C_{c}^{\infty}$ and $0 < ∣ h ∣ < dist (supp η, \partial Ω)$ the class $v : = - δ_{- h}^{k} (η^{2} δ_{h}^{k} u)$ lies in $H_{0}^{1}$ and is an admissible test class in the weak equation. (The difference-quotient test function and its commutators)

[F5]

Scales: for every $x_{0}$ and $0 < r < R$ there is $η \in C_{c}^{\infty} (B_{(r + R) / 2} (x_{0}))$ with $0 \leq η \leq 1$ , $η = 1$ on $B_{r} (x_{0})$ and $∥ D η ∥_{\infty} \leq C (n) / (R - r)$ ; and the scaled Caccioppoli inequality holds on concentric balls $B_{ρ} (x_{0}) ⋐ B_{R} (x_{0})$ . (The standard smooth step function, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Compactly supported scaled Euclidean bumps, Scaled Caccioppoli inequality on concentric balls)

[F6]

Young and Cauchy--Schwarz: $a b \leq ε a^{2} + (4 ε)^{- 1} b^{2}$ for $ε > 0$ and real $a, b \geq 0$ , and $∣ \int g \overline{h} d x ∣ \leq ∥ g ∥_{L^{2}} ∥ h ∥_{L^{2}}$ . (Young's inequality for conjugate real exponents, Holder's inequality for integrals, including the endpoint cases)

[F7]

Difference-quotient characterisation of $W^{1, p}$ for $1 < p < \infty$ : (1) if $u \in W^{1, p} (Ω)$ then $∥ δ_{h}^{i} u ∥_{L^{p} (Ω^{'})} \leq ∥ D_{i} u ∥_{L^{p} (Ω)}$ for $0 < ∣ h ∣ < dist (Ω^{'}, \partial Ω)$ ; (2) conversely, if $∥ δ_{h}^{i} u ∥_{L^{p} (Ω^{'})} \leq C$ for all $0 < ∣ h ∣ < dist (Ω^{'}, \partial Ω) / 2$ , then $D_{i} u \in L^{p} (Ω^{'})$ with $∥ D_{i} u ∥_{L^{p} (Ω^{'})} \leq C$ . The weak-limit supplier proves the same converse from bounds for all $0 < ∣ h ∣ < h_{0}$ for any finite positive $h_{0}$ below the domain margin. (The difference-quotient characterisation of $W^{1, p}$ for $1 < p < \infty$ , Uniformly bounded difference quotients represent a weak derivative)

Proof

technique · direct

1.1F4F5

Setup. Fix $x_{0}$ and $0 < r < R$ , put $ρ : = (r + R) / 2$ and $h_{0} : = (R - r) / 4$ , and choose the real cutoff $η (x) = σ ((s_{*}^{2} - ∣ x - x_{0} ∣^{2}) / (s_{*}^{2} - r^{2}))$ with $s_{*} = (r + ρ) / 2$ and the fixed smooth step of [F5]. Its support lies in ${\overline{B}}_{s_{*}} ⋐ B_{ρ}$ , it equals one on $B_{r}$ , and the chain rule gives $∥ D η ∥_{\infty} \leq 4 ∥ σ^{'} ∥_{\infty} / (ρ - r) = C_{1} / (R - r)$ with $C_{1} = 8 ∥ σ^{'} ∥_{\infty}$ , so that $∥ D (η^{2}) ∥_{\infty} \leq 2 C_{1} / (R - r)$ . For $0 < ∣ h ∣ < h_{0}$ and $k \in {1, \dots, n}$ the class $v : = - δ_{- h}^{k} (η^{2} δ_{h}^{k} u)$ is defined and admissible in the weak equation by [F4], and all difference quotients below are taken on $B_{ρ} (x_{0})$ .

2.1F2F3step 1.1

Constant-coefficient translation. For $w = η^{2} δ_{h}^{k} u$ , its support and all small translates are compactly contained in $B_{R}$ . Discrete integration by parts and commutation of weak derivatives therefore give $a (u, - δ_{- h}^{k} w) = a (δ_{h}^{k} u, w)$ , since the coefficients are constant. This use is confined to the supported test $w$ ; an arbitrary $H_{0}^{1} (B_{R})$ test need not admit a translation staying inside the ball.

3.1F1F2F3F7step 2.1algebra

Expansion and datum. Put $E_{h} = ∥ η δ_{h}^{k} D u ∥_{2}$ and $s = R - r$ . The product rule expands $a (δ_{h}^{k} u, η^{2} δ_{h}^{k} u)$ into the accretive principal term, the principal cutoff term, and drift/reaction terms. On the supported cutoff neighbourhood, with shifts remaining inside $B_{ρ}$ , the coordinate quotient bound gives $∥ δ_{h}^{k} u ∥_{2} \leq ∥ D_{k} u ∥_{L^{2} (B_{ρ})}$ after decreasing $h_{0}$ to the cutoff-support margin if necessary. Thus $∥ v ∥_{2} \leq C (E_{h} + s^{- 1} ∥ D u ∥_{L^{2} (B_{ρ})})$ . This only uses norms on valid shrunken domains, not an undefined quotient on all $B_{R}$ .

4.1F2F6step 3.1algebra

Absorption. The principal cutoff term is at most $C M_{a} s^{- 1} E_{h} ∥ D u ∥_{L^{2} (B_{ρ})}$ . The drift and reaction terms are bounded by $C M_{b} E_{h} ∥ D u ∥_{L^{2} (B_{ρ})} + M_{c} ∥ D u ∥_{L^{2} (B_{ρ})}^{2}$ . The datum pairing is at most $C ∥ f ∥_{L^{2} (B_{R})} (E_{h} + s^{- 1} ∥ D u ∥_{L^{2} (B_{ρ})})$ . Young's inequality and $Re P_{h} \geq θ E_{h}^{2}$ therefore give $E_{h}^{2} \leq C ((1 + s^{- 2}) ∥ D u ∥_{L^{2} (B_{ρ})}^{2} + ∥ f ∥_{L^{2} (B_{R})}^{2})$ , where $C$ depends only on $n, θ, M_{a}, M_{b}, M_{c}$ , uniformly in sufficiently small $h$ .

5.1F1F2F5F6step 4.1algebra

Refined gradient estimate. To eliminate the intermediate gradient, choose a smooth cutoff $β$ equal to one on $B_{ρ}$ , supported in $B_{R}$ with $∥ D β ∥_{\infty} \leq C / s$ , and test the weak equation with $β^{2} u$ . The Caccioppoli computation behind [F5], taking real parts and absorbing the principal cutoff and drift products by Young, gives $∥ D u ∥_{L^{2} (B_{ρ})}^{2} \leq C ((1 + s^{- 2}) ∥ u ∥_{L^{2} (B_{R})}^{2} + ∥ f ∥_{L^{2} (B_{R})} ∥ u ∥_{L^{2} (B_{R})})$ . Set $t = (1 + s^{- 2})^{- 1}$ and use $∥ f ∥_{2} ∥ u ∥_{2} \leq t ∥ f ∥_{2}^{2} + (4 t)^{- 1} ∥ u ∥_{2}^{2}$ . Then $∥ D u ∥_{L^{2} (B_{ρ})}^{2} \leq C ((1 + s^{- 2}) ∥ u ∥_{2}^{2} + (1 + s^{- 2})^{- 1} ∥ f ∥_{2}^{2})$ . Substituting this bound into step 4.1 yields $E_{h}^{2} \leq C ((1 + s^{- 2})^{2} ∥ u ∥_{2}^{2} + ∥ f ∥_{2}^{2})$ . This retains the forcing coefficient at every scale.

6.1step 5.1F7algebra∎

Conclusion. Since $η = 1$ on $B_{r}$ , step 5.1 bounds every coordinate quotient $δ_{h}^{k} D_{j} u$ on $B_{r}$ uniformly for all sufficiently small $h$ . The converse criterion [F7], applied with any finite threshold below both this support margin and $(R - r) / 2$ , gives each $D_{k} D_{j} u \in L^{2} (B_{r})$ with the same bound. Summing the finitely many second-derivative bounds and taking square roots gives the displayed estimate $∥ D^{2} u ∥_{L^{2} (B_{r})} \leq C ((1 + (R - r)^{- 2}) ∥ u ∥_{L^{2} (B_{R})} + ∥ f ∥_{L^{2} (B_{R})})$ .

Source notes

Laugesen's Theorem 5.6 (printed pp. 108-110) proves the estimate for $L = - Δ$ by difference quotients with the cutoff test function, and Hunter's Theorem 4.27 (printed pp. 110-114) carries out the same scheme for general divergence-form operators; the constant-coefficient case has no coefficient commutators, so the error terms in step 3.1 contain only the cutoff gradients $D_{i} (η^{2})$ , which is why the step-size $h$ disappears from the final constant. The scaled Caccioppoli inequality supplies the $(R - r)^{- 2} ∥ u ∥_{L^{2} (B_{R})}$ term exactly at the scale of the statement.

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Sources

Richard S. Laugesen, Linear Analysis and Partial Differential Equations (University of Illinois, 2020, complete 158-page graduate notes) (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)

Interior H2H2 estimate for constant-coefficient elliptic equations