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A finite partition glues the local interior and boundary $H^{2}$ estimates

Statement

Assume Countable Choice. Let $Ω \subset R^{n}$ be a bounded $C^{2}$ domain (Bounded C^k domains and boundary charts), $n \geq 2$ , let $L, a$ be as in Uniformly elliptic divergence-form operators and their sesquilinear forms with $a^{i j} \in W^{1, \infty} (Ω)$ , $b^{i}, c \in L^{\infty} (Ω)$ , and let $f \in L^{2} (Ω)$ . Suppose $u \in H^{1} (Ω)$ is a local weak solution of $L u = f$ on $Ω$ (Local weak solutions of a divergence-form operator) and fix a finite ambient smooth partition of unity near $\overline{Ω}$ whose pieces are supported in interior patches compactly contained in $Ω$ or in compact ambient boundary-chart patches $(W_{ℓ}, Φ_{ℓ})$ . Assume every boundary piece has the quantitative bound $∥ \hat{ζ_{ℓ} u} ∥_{H^{2} (Q_{ℓ})} \leq C_{ℓ} (∥ f ∥_{L^{2} (Ω)} + ∥ u ∥_{L^{2} (Ω)}),$ where the flattened half-patch $Q_{ℓ}$ contains its entire support, and the chart/inverse derivatives through order two and Jacobians have fixed uniform bounds there. These are hypotheses, rather than consequences of an unspecified boundary condition on $u$ . Then $u \in H^{2} (Ω)$ and there is $C$ , depending only on $n, θ$ , the coefficient bounds, the fixed partition/chart bounds and the constants $C_{ℓ}$ , with $∥ u ∥_{H^{2} (Ω)} \leq C (∥ f ∥_{L^{2} (Ω)} + ∥ u ∥_{L^{2} (Ω)}) .$ The proof uses a finite smooth partition of unity subordinate to the cover, the localisation identities of Localisation of a weak solution up to a bounded first-order term, and the finiteness of the cover; no choice of a cover beyond the finite chart neighbourhoods supplied by the definition is used.

Facts & Assumptions

Given: Countable Choice; the bounded $C^{2}$ domain and its finite boundary atlas; the coefficient package; the solution $u$ ; and the stated local $H^{2}$ bounds on the interior set and the flattened localisations.

[F1]

Localisation identity: for an ambient smooth cutoff $ζ$ supported in a chart neighbourhood, the localized weak equation is $L (ζ u) = ζ f - (D_{i} ζ) a^{i j} D_{j} u - D_{i} (a^{i j} u D_{j} ζ) + b^{i} (D_{i} ζ) u .$ Expanding the divergence gives an $L^{2}$ datum whose norm is bounded by $C (ζ) (∥ f ∥_{2} + ∥ u ∥_{H^{1}})$ , since $a \in W^{1, \infty}$ and $b, c$ are bounded. This bound alone does not replace $∥ u ∥_{H^{1}}$ by $∥ u ∥_{2}$ ; that replacement must come from the assumed quantitative local bounds or, in a zero-trace application, a separate energy estimate. (Local weak solutions of a divergence-form operator, Localisation of a weak solution up to a bounded first-order term)

[F2]

A finite ambient smooth partition can be chosen subordinate to a finite cover of $\overline{Ω}$ by an interior region and boundary chart neighbourhoods, with cutoffs supported in compactly contained ambient patches. The interior region may be enlarged inside $Ω$ to cover the compact set remaining outside the boundary patches. (Finite ambient partitions near compact sets, Compactly supported scaled Euclidean bumps, Bounded C^k domains and boundary charts)

[F3]

On every pair of interior balls $B_{r} (x) ⋐ B_{R} (x) ⋐ Ω$ , the interior $H^{2}$ theorem gives a bound for $u$ on $B_{r} (x)$ by $C (∥ f ∥_{L^{2} (B_{R} (x))} + ∥ u ∥_{L^{2} (B_{R} (x))})$ . On each boundary chart, the Statement assumes the corresponding quantitative $H^{2}$ bound for the flattened localization, obtained from the tangential and normal estimates. The constants depend on the fixed balls or chart, cutoffs and coefficient bounds; these are local estimates for the gluing step, not consequences of a boundary condition on a general local weak solution. (Interior $H^{2}$ regularity for divergence-form equations, Tangential $H^{2}$ estimate near a flat Dirichlet boundary, The normal second derivative is recovered from the equation)

[F4]

On compactly contained ambient $C^{2}$ chart patches, change of variables and the weak chain rule transport $H^{2}$ norms in both directions with uniform constants: second derivatives use only first and second chart derivatives and derivatives of the function through order two. The compact ambient bounds remain uniform on half-patches reaching the boundary. (Weak divergence-form equations are invariant under $C^{2}$ boundary charts, $C^{2}$ flattening preserves uniform ellipticity quantitatively, Bounded C^k domains and boundary charts)

Proof

1.1F2F3given

Use the finite partition fixed in the Statement. Compactness and the graph definition permit such a partition: finitely many boundary patches cover $\partial Ω$ , their complement in $\overline{Ω}$ is compact in $Ω$ , and finitely many interior balls cover it; [F2] supplies the subordinate ambient smooth functions. The boundary estimates assumed in the Statement concern these actual fixed pieces and their whole supports, so no new unestimated boundary localization is substituted.

2.1F1step 1.1

Localized equations. Each $ζ_{j} u$ belongs to $H^{1} (Ω)$ by the product rule and has the datum in [F1]. Ambient cutoffs are admissible even at boundary patches: their restrictions multiply the Sobolev class, and the distributional identity is tested on compact subsets of $Ω$ . The extra terms are bounded by $C_{j} (∥ f ∥_{2} + ∥ u ∥_{H^{1}})$ . The sharper $L^{2}$ -based local estimates consumed below are precisely those assumed in [F3]; no zero-trace condition is inferred for a general $u \in H^{1} (Ω)$ .

3.1F3step 1.1step 2.1

For an interior piece, choose nested compactly interior open sets containing its support. The interior theorem in [F3] bounds $u$ in $H^{2}$ on the inner neighbourhood by $C (∥ f ∥_{2} + ∥ u ∥_{2})$ . The smooth multiplier rule bounds the piece there; its cutoff support is compact in $Ω$ , so its weak derivatives extend by zero across the artificial edges inside $Ω$ . Each such piece therefore has the required $H^{2} (Ω)$ bound.

3.2F3F4step 1.1step 2.1

Boundary pieces. Each assumed flattened $H^{2}$ estimate in [F3] holds on a half-patch containing the entire support of the corresponding cutoff. The compact ambient chart bounds and [F4] transport it back to $∥ ζ_{ℓ} u ∥_{H^{2} (Ω \cap W_{ℓ})} \leq C_{ℓ}^{'} (∥ f ∥_{2} + ∥ u ∥_{2})$ . The cutoff vanishes near the artificial chart edges, so the local derivatives extend by zero inside $Ω$ and give the same $H^{2} (Ω)$ bound. No extension across the actual boundary of $Ω$ is required.

4.1step 3.1step 3.2algebra

Summing. Since $u = \sum_{j = 0}^{m} ζ_{j} u$ almost everywhere and each piece belongs to $H^{2} (Ω)$ , linearity of weak derivatives gives $u \in H^{2} (Ω)$ and $∥ u ∥_{H^{2} (Ω)} \leq \sum_{j} ∥ ζ_{j} u ∥_{H^{2} (Ω)} \leq C (∥ f ∥_{2} + ∥ u ∥_{2})$ . The finite sum of local constants depends on the fixed atlas, cutoffs and coefficient data, as asserted.

5.1step 4.1∎

Conclusion. The given quantitative interior and boundary estimates glue to the displayed global estimate. The PDE estimates supply the local hypotheses in Dirichlet applications; the finite partition argument itself adds no boundary condition or additional estimate for the localized forcing.

Source notes

Hunter's proof of Theorem 4.30 (printed p. 115) reduces the global statement to the half-space case by a partition of unity and a flattening of the boundary; Simon's Lecture 9 (printed pp. 88-90) performs the same reduction. The lemma records the reduction step separately so that the flat-boundary estimates can be consumed by the global Dirichlet theorem.

Depends on

Used by

Global H² Dirichlet regularity Theorem

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)
Leon Simon, Lectures on Partial Differential Equations (Stanford, complete 223-page author scan) (standard reference, not scraped)

A finite partition glues the local interior and boundary H2H2 estimates