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The normal second derivative is recovered from the equation

Statement

Assume Countable Choice. Let $n \geq 1$ and give $R^{n}$ its Euclidean metric ( $R^{n}$ as the set of functions $n \to R$ , and $d_{1}$ , $d_{2}$ , $d_{\infty}$ are metrics on it). In this item relabel its zero-based coordinates by $x_{i} : = x (i - 1)$ for $1 \leq i \leq n$ ; $D_{i}$ and $e_{i}$ refer to these coordinates. Define the open half-space $H : = {x \in R^{n} : x_{n} > 0}$ and its boundary hyperplane $\partial H = {x \in R^{n} : x_{n} = 0}$ . For $r > 0$ , write $B_{r} (x_{0}) : = {x : ∣ x - x_{0} ∣ < r}$ (Open ball, closed ball and sphere in a metric space), so the boundary half-ball is $Q = {x : ∣ x - x_{0} ∣ < r, x_{n} > 0}$ with $(x_{0})_{n} = 0$ . Let $u \in H_{0}^{1} (H) \cap H_{l o c}^{2} (H)$ be such that the tangential second derivatives $D_{k} D_{i} u$ ( $k < n$ ) and the derivatives $D_{n} D_{i} u$ for $i < n$ exist in $L_{l o c}^{2} (H)$ ; suppose $u$ solves $L u = f$ weakly on $H$ with $a^{i j} \in W^{1, \infty} (H)$ uniformly elliptic with constant $θ$ , $b^{i}, c \in L^{\infty} (H)$ and $f \in L_{l o c}^{2} (H)$ (Local weak solutions of a divergence-form operator). Then the missing normal derivative $D_{n} D_{n} u$ exists in $L_{l o c}^{2} (H)$ and satisfies throughout $H$ , in the almost-everywhere strong form, $D_{n} D_{n} u = \frac{1}{a^{n n}} (- \sum_{(i, j) \neq (n, n)} a^{i j} D_{i} D_{j} u - \sum_{i, j} (D_{i} a^{i j}) D_{j} u + b^{i} D_{i} u + c u - f)$ almost everywhere, with the pointwise bound $∣ D_{n} D_{n} u ∣ \leq θ^{- 1} (M_{a} \sum_{(i, j) \neq (n, n)} ∣ D_{i} D_{j} u ∣ + C (n) (M_{1} + M_{b}) ∣ D u ∣ + M_{c} ∣ u ∣ + ∣ f ∣)$ and the corresponding $L^{2}$ estimate on each boundary half-ball $Q = B_{r} (x_{0}) \cap H$ , $x_{0} \in \partial H$ , on which $f$ and all the nonnormal second derivatives on the right are in $L^{2} (Q)$ . In particular those hypotheses imply $D_{n}^{2} u \in L^{2} (Q)$ , with $∥ D_{n}^{2} u ∥_{L^{2} (Q)} \leq C (\sum_{(i, j) \neq (n, n)} ∥ D_{i} D_{j} u ∥_{L^{2} (Q)} + ∥ D u ∥_{L^{2} (Q)} + ∥ u ∥_{L^{2} (Q)} + ∥ f ∥_{L^{2} (Q)}) .$ Uniform ellipticity gives $Re a^{n n} \geq θ$ and hence $∣ a^{n n} ∣ \geq θ$ , which makes division legitimate for real or complex coefficients.

Facts & Assumptions

Given: Countable Choice; the half-space; the coefficient package; the data and the solution with the stated partial second derivatives.

[F1]

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

[F2]

Coefficient package: $∣ a^{i j} ∣ \leq M_{a}$ , $∣ D_{i} a^{i j} ∣ \leq M_{1}$ , $∣ b^{i} ∣ \leq M_{b}$ , $∣ c ∣ \leq M_{c}$ a.e. and $Re (a^{i j} ξ_{j} \overline{ξ_{i}}) \geq θ ∣ ξ ∣^{2}$ ; in particular $Re a^{n n} \geq θ > 0$ and $∣ a^{n n} ∣ \geq θ$ a.e. (Uniformly elliptic divergence-form operators and their sesquilinear forms)

[F3]

On an open set where $u$ is a strong solution, integration by parts in [F1] against $φ \in C_{c}^{\infty}$ gives $\int_{H} (- D_{i} (a^{i j} D_{j} u) + b^{i} D_{i} u + c u - f) \overline{φ} d x = 0$ for every test function, where $D_{i} (a^{i j} D_{j} u) = (D_{i} a^{i j}) D_{j} u + a^{i j} D_{i} D_{j} u$ with the displayed second derivatives integrable; since the bracket lies in $L_{l o c}^{2}$ and is orthogonal to every $C_{c}^{\infty}$ function, it vanishes a.e. (The notation $H^{k}$ and the reserved zero-boundary symbol, Smooth compactly supported functions of an open set are dense in $L^{2}$ )

Proof

technique · direct

1.1F1F2F3

Strong identity in the open half-space. On every compactly contained open subset of $H$ , the hypothesis $u \in H_{l o c}^{2} (H)$ and the multiplier rule of The cutoff difference-quotient commutator estimate for $a \in W^{1, \infty}$ give $D_{i} (a^{i j} D_{j} u) = (D_{i} a^{i j}) D_{j} u + a^{i j} D_{i} D_{j} u$ in $L^{2}$ . The weak equation and density of smooth tests imply $- D_{i} (a^{i j} D_{j} u) + b^{i} D_{i} u + c u = f$ almost everywhere there. A countable exhaustion proves this identity almost everywhere throughout $H$ ; no boundary $H^{2}$ regularity has been assumed.

2.1step 1.1F2algebra

Algebraic recovery. Separating the $(n, n)$ term gives $a^{n n} D_{n}^{2} u = - \sum_{(i, j) \neq (n, n)} a^{i j} D_{i} D_{j} u - \sum_{i, j} (D_{i} a^{i j}) D_{j} u + b^{i} D_{i} u + c u - f$ . Testing ellipticity with $ξ = e_{n}$ gives $Re a^{n n} \geq θ$ , hence $∣ 1 / a^{n n} ∣ \leq θ^{- 1}$ even for complex coefficients. Division therefore gives the displayed formula and pointwise bound almost everywhere on $H$ .

3.1step 2.1F2algebra

Estimate up to the flat boundary. Let $Q = B_{r} (x_{0}) \cap H$ be a boundary half-ball satisfying the integrability conditions in the Statement. The right-hand side of step 2.1 belongs to $L^{2} (Q)$ because the nonnormal derivatives and $f$ do, and $u \in H^{1} (H)$ . Integrating the pointwise bound over $Q$ and using the triangle inequality proves the displayed $L^{2} (Q)$ estimate. The already existing interior weak derivative $D_{n}^{2} u$ equals this $L^{2} (Q)$ function on every compact test support in $Q$ , so the same function represents that weak derivative on the entire open half-ball. This recovers boundary integrability without first assuming $u \in H^{2} (Q)$ .

4.1step 2.1step 3.1∎

Conclusion. The equation determines the normal derivative throughout $H$ and bounds it on every boundary half-ball where the tangential and mixed second derivatives and forcing have been controlled. Together with the tangential difference-quotient estimate, this supplies the missing second derivative up to the flat boundary for real or complex coefficients.

Source notes

Hunter (printed pp. 115-116) recovers $\partial_{n}^{2} u$ from the equation after the tangential second derivatives have been estimated, and Teschl's Lemma 10.18 (printed p. 242) proceeds in the same order. The formula is the algebraic solve for $a^{n n} D_{n} D_{n} u$ in the strong form of the equation; the ellipticity bound $∣ a^{n n} ∣ \geq θ$ obtained from $ξ = e_{n}$ is what makes the division legitimate.

Depends on

Used by

Dependency tree · two levels

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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)