Higher-order boundary regularity for Dirichlet problems

The induction claim is ${\mathrm{P}}_j$: under the hypotheses of the Statement with $k=j$, the solution satisfies $u\in H^{j+2}(\mathrm{\Omega })$ with $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{j+2}(\mathrm{\Omega })}\le C_j(\mathrm{\parallel }f{\mathrm{\parallel }}_{H^j(\mathrm{\Omega })}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })})$, where $C_j$ depends only on $n,j,\mathrm{\Omega }$, principal coefficient bounds through order $j+1$, and lower-order coefficient bounds through order $j$.

Base case $j=0$. This is [F4] verbatim.

Assume ${\mathrm{P}}_{j-1}$ with $1\le j\le k$, so $u\in H^{j+1}(\mathrm{\Omega })$ with the induction bound. On each fixed ambient $C^{j+2}$ boundary patch, [F5] transports this regularity and the original equation to a half-patch. The formulas for the transformed matrix use the Jacobian and two first chart derivatives; differentiating them through order $j+1$ uses only original coefficient derivatives through order $j+1$ and chart/inverse derivatives through order $j+2$. Thus the transformed principal matrix is $W^{j+1,\mathrm{\infty }}$, the drift and reaction are $W^{j,\mathrm{\infty }}$, and the forcing is $H^j$ with bounded norms. Choose a real ambient cutoff $\eta$ vanishing near the artificial edges and equal to one on a smaller half-ball. For $z=\eta \widehat{u}$, the expanded localization formula has a datum $g\in H^j$ with $\mathrm{\parallel }g{\mathrm{\parallel }}_{H^j}\le C(\mathrm{\parallel }f{\mathrm{\parallel }}_{H^j(\mathrm{\Omega })}+\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{j+1}(\mathrm{\Omega })})$: its terms use derivatives of $\widehat{u}$ through order $j+1$, principal coefficients through order $j+1$, and $\eta$ through order $j+2$. The zero-trace transfer of the chart lemma gives $z\in H_0^1(H)$ after zero extension at artificial edges, and $z\in H^{j+1}(H)$ by [F5]. Extend the coefficients to $H$ using a larger ambient cutoff, equal to one near the support of $z$, and a constant positive identity matrix outside the patch, as in the base theorem; this preserves ellipticity and all stated Sobolev bounds and leaves $Lz=g$.

Zero-trace tangential derivatives and their estimates. If $w\in H^2(H)\cap H_0^1(H)$ and $\mathrm{\ell }<n$, then ${\delta }_h^{\mathrm{\ell }}w\in H_0^1(H)$ by the bounded fixed-$h$ shift operations in [F1]. Applying the tangential strong convergence of [F5] to $w$ and every $D_{i}w\in H^1(H)$ gives ${\delta }_h^{\mathrm{\ell }}w\to D_{\mathrm{\ell }}w$ in $H^1(H)$; closedness of $H_0^1(H)$ therefore gives $D_{\mathrm{\ell }}w\in H_0^1(H)$. Iterating for $z\in H^{j+1}(H)$ proves $w_{\alpha }=D_{\mathrm{t}\mathrm{a}\mathrm{n}}^{\alpha }z\in H_0^1(H)$ for $\mathrm{\mid }\alpha \mathrm{\mid }=j$. The differentiated equation [F2] and the multiplier rule give its datum $h_{\alpha }\in L^2(H)$ with $\mathrm{\parallel }{h}_{\alpha }{\mathrm{\parallel }}_2\le C(\mathrm{\parallel }g{\mathrm{\parallel }}_{H^j(H)}+\mathrm{\parallel }z{\mathrm{\parallel }}_{H^{j+1}(H)})$. After scaling the fixed supports into the outer half-ball, the tangential estimate in [F1] applies to $w_{\alpha }$, and the interior H2 theorem from [F3] supplies its $H_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(H)$ regularity. The normal-recovery estimate in [F1] then yields its full H2 bound on the smaller half-ball. Varying $\alpha$ controls all order-$j+2$ derivatives of $z$ with at most two normal factors.

For the remaining derivatives, the interior estimate [F3] already gives $z\in H_{\mathrm{l}\mathrm{o}\mathrm{c}}^{j+2}(H)$, so differentiate the expanded strong equation $-a^{pq}{D}_p{D}_{q}z-(D_p{a}^{pq})D_{q}z+b^p{D}_{p}z+cz=g$ on compact interior subsets using the proved multiplier rule. For a multi-index $\gamma$ of order $j$ with ${\gamma }_n=r-2$, the sole term with $r$ normal factors is $-a^{nn}{D}_n^2{D}^{\gamma }z$. Every other order-$j+2$ term has at most $r-1$ normal factors; terms where a derivative hits a coefficient use only derivatives of $z$ through order $j+1$, principal coefficients through order $j+1$ and lower-order coefficients through order $j$. The forcing derivative $D^{\gamma }g$ is $L^2$. Starting with the at-most-two-normal derivatives from step 4.1, induction on $r$ and $\mathrm{\mid }{a}^{nn}\mathrm{\mid }\ge {\theta }_0>0$ bound every remaining derivative in $L^2$ on the smaller boundary half-ball. The a.e. identities hold throughout it by a countable exhaustion of its interior; every compact test support lies in that interior, so these $L^2$ fields represent the global weak derivatives on the open half-ball. No multiplication of an undefined distribution by a merely Lipschitz reciprocal is required.

Completing the induction. Steps 3.1--5.1, using the boundary, differentiated-equation, interior, and base estimates of [F1]–[F4], bound every weak derivative of order $j+2$ on the chart-localised regions and the interior region; summing the finitely many local bounds and gluing with the partition of unity gives $u\in H^{j+2}(\mathrm{\Omega })$ with $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{j+2}(\mathrm{\Omega })}\le C_j(\mathrm{\parallel }f{\mathrm{\parallel }}_{H^j(\mathrm{\Omega })}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })})$, which is ${\mathrm{P}}_j$; by induction ${\mathrm{P}}_k$ holds.

Conclusion. Under $C^{k+2}$ boundary regularity, coefficients of order $k+1$ for the principal part and order $k$ for the lower-order terms, and data in $H^k$, the zero-trace Dirichlet solution lies in $H^{k+2}(\mathrm{\Omega })$ with the displayed two-derivative gain.