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Tangential $H^{2}$ estimate near a flat Dirichlet boundary

Statement

Assume Countable Choice. Let $H = {x_{n} > 0}$ be the upper half-space, $n \geq 1$ , $K \in {R, C}$ , let $L, a$ be as in Uniformly elliptic divergence-form operators and their sesquilinear forms with $a^{i j} \in W^{1, \infty} (H)$ , $b^{i}, c \in L^{\infty} (H)$ and bounds $θ, M_{a}, M_{b}, M_{c}, M_{1}$ , let $f \in L^{2} (H)$ , and let $u \in H_{0}^{1} (H)$ be supported in $B_{1} (0) \cap \overline{H}$ and solve $L u = f$ weakly on $H$ (Local weak solutions of a divergence-form operator). Then for every tangential index $k < n$ and every $i$ the weak derivative $D_{k} D_{i} u$ belongs to $L^{2} (B_{1 / 2} (0) \cap H)$ with $\sum_{k < n} \sum_{i = 1}^{n} \int_{B_{1 / 2} (0) \cap H} ∣ D_{k} D_{i} u ∣^{2} d x \leq C (∥ f ∥_{L^{2} (H)}^{2} + ∥ u ∥_{L^{2} (H)}^{2}),$ where $C = C (n, θ, M_{a}, M_{b}, M_{c}, M_{1})$ is independent of the step size. Only tangential difference quotients of $u$ are used, so no extension of $u$ across the boundary is invoked.

Facts & Assumptions

Given: Countable Choice; the half-space $H$ ; the coefficients and their bounds; the datum $f \in L^{2} (H)$ ; and the local weak solution $u \in H_{0}^{1} (H)$ supported in $B_{1} (0) \cap \overline{H}$ .

[F1]

Local weak solution and Dirichlet test class: $a (u, φ) = \int_{H} f \overline{φ} d x$ for every $φ \in C_{c}^{\infty} (H)$ , and products of $u$ with functions of $C_{c}^{\infty} (R^{n})$ lie in $H_{0}^{1} (H)$ . (Local weak solutions of a divergence-form operator, the explicitly defined half-space $H = {x_{n} > 0}$ )

[F2]

Coefficient package: $∣ a^{i j} ∣ \leq M_{a}$ , $∣ D_{k} a^{i j} ∣ \leq M_{1}$ , $∣ b^{i} ∣ \leq M_{b}$ , $∣ c ∣ \leq M_{c}$ a.e. and $Re (a^{i j} ξ_{j} \overline{ξ_{i}}) \geq θ ∣ ξ ∣^{2}$ . (Uniformly elliptic divergence-form operators and their sesquilinear forms)

[F3]

Tangential test class and principal pairing: for $k < n$ and a real cutoff $η \in C_{c}^{\infty} (R^{n})$ , $v = - δ_{- h}^{k} (η^{2} δ_{h}^{k} u)$ lies in $H_{0}^{1} (H)$ . The weak identity extends from smooth tests to this class by density and boundedness of the form. Difference-quotient integration by parts and the product identity give $\int_{H} a^{i j} D_{j} u \overline{D_{i} v} = \int_{H} η^{2} a^{i j} (x + h e_{k}) δ_{h}^{k} D_{j} u \overline{δ_{h}^{k} D_{i} u} + R_{a} .$ Here $R_{a}$ consists of the principal coefficient quotient and cutoff terms only. Writing $E_{h} = ∥ η δ_{h}^{k} D u ∥_{2}$ , these satisfy $∣ R_{a} ∣ \leq ε E_{h}^{2} + C_{ε} ∥ D u ∥_{2}^{2}$ . No difference quotient of $b$ or $c$ is used. (The difference-quotient test function and its commutators, Difference-quotient calculus: integration by parts, product rule, commutation, Young's inequality for conjugate real exponents)

[F4]

Difference-quotient calculus and characterisation: difference quotients commute with weak derivatives, and the characterisation of $W^{1, p}$ for $1 < p < \infty$ holds: a uniform bound $∥ δ_{h}^{k} w ∥_{L^{2} (Ω^{'})} \leq C$ for $0 < ∣ h ∣ < h_{0}$ implies $D_{k} w \in L^{2} (Ω^{'})$ with $∥ D_{k} w ∥_{L^{2} (Ω^{'})} \leq C$ , whenever $δ_{h}^{k} w$ is defined on $Ω^{'}$ for those $h$ ; for tangential directions and $Ω^{'} = B_{1 / 2} (0) \cap H$ this validity holds for all $h$ . (Difference-quotient calculus: integration by parts, product rule, commutation, The difference-quotient characterisation of $W^{1, p}$ for $1 < p < \infty$ , Uniformly bounded difference quotients represent a weak derivative)

[F5]

Young and Cauchy--Schwarz inequalities with a free $ε > 0$ , and the elementary bound $∥ δ_{h}^{k} w ∥_{L^{2}} \leq ∥ D_{k} w ∥_{L^{2}}$ for $w \in H^{1}$ . (Young's inequality for conjugate real exponents, Holder's inequality for integrals, including the endpoint cases, The difference-quotient characterisation of $W^{1, p}$ for $1 < p < \infty$ )

[F6]

Fix a real smooth bump equal to one on $B_{1 / 2}$ and supported in $B_{1}$ ; its gradient has a finite bound depending only on this fixed choice and the dimension. (A smooth bump between concentric Euclidean balls)

Proof

1.1F1F3F6

Setup. Choose $η \in C_{c}^{\infty} (B_{1} (0))$ with $0 \leq η \leq 1$ , $η = 1$ on $B_{1 / 2} (0)$ and $∥ D η ∥_{\infty} \leq C_{1} = C_{1} (n)$ as in [F6]; fix a tangential index $k < n$ and $0 < ∣ h ∣ < 1 / 2$ . Since the shift is tangential, $δ_{h}^{k} u$ and $δ_{- h}^{k} (η^{2} δ_{h}^{k} u)$ are defined on $B_{1 / 2} (0) \cap H$ and $v$ is an admissible test class by [F3].

1.2F1F2F5

Global gradient bound. Since $u \in H_{0}^{1} (H)$ and $f \in L^{2} (H)$ , density permits $u$ itself as a test. Taking real parts gives $θ ∥ D u ∥_{2}^{2} \leq ∥ f ∥_{2} ∥ u ∥_{2} + C (n) M_{b} ∥ D u ∥_{2} ∥ u ∥_{2} + M_{c} ∥ u ∥_{2}^{2}$ . Young's inequality absorbs half the gradient term and yields $∥ D u ∥_{2}^{2} \leq C (∥ f ∥_{2}^{2} + ∥ u ∥_{2}^{2})$ . This controls the entire half-space gradient and does not rely on a cutoff equal to one on the support of $u$ .

2.1F2F3step 1.1

Principal pairing and ellipticity. By [F3], the principal pairing is $P_{h} + R_{a}$ , where $P_{h} = \int_{H} η^{2} a^{i j} (x + h e_{k}) δ_{h}^{k} D_{j} u \overline{δ_{h}^{k} D_{i} u}$ . Tangential translation preserves $H$ , so $Re P_{h} \geq θ E_{h}^{2}$ by [F2]. The remainder is bounded by $ε E_{h}^{2} + C_{ε} ∥ D u ∥_{2}^{2}$ uniformly for small $h$ by [F3].

3.1F1F2F5step 2.1algebra

Datum and lower-order terms. The difference-quotient bound and the product rule give $∥ v ∥_{2} \leq ∥ D_{k} (η^{2} δ_{h}^{k} u) ∥_{2} \leq C (η) (E_{h} + ∥ D u ∥_{2})$ . Thus the weak equation, with the lower-order terms left undifferentiated, bounds $∣ \int_{H} (f - b^{i} D_{i} u - c u) \overline{v} ∣$ by $C (∥ f ∥_{2} + M_{b} ∥ D u ∥_{2} + M_{c} ∥ u ∥_{2}) (E_{h} + ∥ D u ∥_{2})$ . Young's inequality gives $ε E_{h}^{2} + C_{ε} (∥ f ∥_{2}^{2} + ∥ D u ∥_{2}^{2} + ∥ u ∥_{2}^{2})$ . Boundedness of $b, c$ is sufficient.

4.1step 2.1step 3.1algebra

Absorption. Combining steps 2.1 and 3.1 and choosing $ε$ small gives $\int_{H} η^{2} ∣ δ_{h}^{k} D u ∣^{2} d x \leq C (∥ f ∥_{L^{2} (H)}^{2} + ∥ u ∥_{L^{2} (H)}^{2} + ∥ D u ∥_{L^{2} (H)}^{2})$ with $C = C (n, θ, M_{a}, M_{b}, M_{c}, M_{1})$ , uniformly in $0 < ∣ h ∣ < 1 / 2$ .

5.1step 4.1step 1.2F4algebra∎

Conclusion. Substituting step 1.2 into step 4.1 and using $η = 1$ on $B_{1 / 2} (0)$ yields $∥ δ_{h}^{k} D_{i} u ∥_{L^{2} (B_{1 / 2} (0) \cap H)} \leq C^{1 / 2} (∥ f ∥_{L^{2} (H)} + ∥ u ∥_{L^{2} (H)})$ for every tangential $k$ and every $i$ , uniformly in $0 < ∣ h ∣ < 1 / 2$ ; [F4] applies with $w = D_{i} u \in L^{2} (H)$ and the tangential validity noted there, so $D_{k} D_{i} u \in L^{2} (B_{1 / 2} (0) \cap H)$ with the same bound; summing over the finitely many $k < n$ and $i \leq n$ gives the displayed estimate.

Source notes

Hunter's proof of Theorem 4.30 (printed p. 115) uses exactly the tangential test function $v = - D_{- h}^{k} (η^{2} D_{h}^{k} u)$ and notes that the zero trace makes it admissible; the same argument as the interior estimate then gives the tangential second derivatives. The global energy test in step 1.2 is valid by density because $u \in H_{0}^{1} (H)$ ; it eliminates the gradient term before the final difference-quotient characterization. The scaffold's scheme is reproduced; no reflection across the boundary is used, and the constant is independent of $h$ .

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Sources

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)
Gerald Teschl, Partial Differential Equations: From Classical to Modern (2025 archived author manuscript, complete 392 pages) (standard reference, not scraped)

Tangential H2H2 estimate near a flat Dirichlet boundary