Global $H^2$ Dirichlet regularity

Given: Countable Choice; the bounded $C^2$ domain and its finite $C^2$ boundary atlas; the coefficient package; the datum $f\in L^2(\mathrm{\Omega })$; and the zero-trace weak solution $u\in H_0^1(\mathrm{\Omega })$.

Weak Dirichlet solution: $a(u,v)={\int }_{\mathrm{\Omega }}f\bar{v}dx$ for every $v\in H_0^1(\mathrm{\Omega })$, and $u\in H_0^1(\mathrm{\Omega })$ is the closure of the $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ classes in $H^1$. (Weak Dirichlet solutions for a divergence-form operator)

Local interior regularity with localization. If $v$ solves the divergence-form equation with $L^2$ datum $g$ on a neighbourhood of $U_1$, the interior $H^2$ theorem bounds $\mathrm{\parallel }v{\mathrm{\parallel }}_{H^2(U_0)}$ by $C(\mathrm{\parallel }g{\mathrm{\parallel }}_{L^2(U_1)}+\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2(U_1)})$ for $U_0⋐U_1$. For a cutoff $\zeta \in C_c^{\mathrm{\infty }}(U_1)$, the product $v=\zeta u$ satisfies such an equation with datum $g_{\zeta }=\zeta f-(D_i\zeta )a^{ij}{D}_{j}u-D_i(a^{ij}u{D}_j\zeta )+b^{i}u{D}_i\zeta ,$ whose $L^2$ norm is bounded by $C_{\zeta }(\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2(U_1)}+\mathrm{\parallel }Du{\mathrm{\parallel }}_{L^2(U_1)}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(U_1)})$ because $a^{ij}\in W^{1,\mathrm{\infty }}$ and $b,c\in L^{\mathrm{\infty }}$ (Interior $H^2$ regularity for divergence-form equations, Uniformly elliptic divergence-form operators and their sesquilinear forms).

Boundary-patch reduction. For each compactly supported boundary localization $\zeta u$, choose an ambient $C^2$ chart with $\mathrm{\Phi }(W\cap \mathrm{\Omega })=V\cap H$. On the compact chart support, $D\mathrm{\Phi },D\psi ,D^2\mathrm{\Phi }$ and the Jacobians are bounded; the chart lemma preserves the weak equation and zero trace, while the flattening lemma gives an accretive $W^{1,\mathrm{\infty }}$ principal matrix with a positive ellipticity constant. The transformed lower-order coefficients are bounded and the transformed localized datum is in $L^2$, with its norm controlled by $\mathrm{\parallel }f{\mathrm{\parallel }}_2+\mathrm{\parallel }Du{\mathrm{\parallel }}_2+\mathrm{\parallel }u{\mathrm{\parallel }}_2$. Extend the transformed principal matrix to all of $H$ by $\chi \tilde{A}+(1-\chi ){\theta }_{0}I$, where $\chi$ is a smooth ambient cutoff equal to one on the support and ${\theta }_0>0$ is the transformed ellipticity constant; extend lower-order coefficients and the datum by multiplication by $\chi$. This preserves uniform ellipticity, the $W^{1,\mathrm{\infty }}$ principal bounds, and the equation for the zero-extended localized solution. After translation and dilation, choose the partition support inside the estimated half-ball $B_{1\mathrm{/}2}\cap H$ while the extended solution is supported in $B_1\cap \bar{H}$. The tangential estimate bounds all tangential second derivatives there; the interior theorem supplies $H_{\mathrm{l}\mathrm{o}\mathrm{c}}^2(H)$ and the normal-recovery lemma, using $\mathrm{Re}{\tilde{a}}^{nn}\ge {\theta }_0$, bounds the remaining derivative. The compact chart bounds transport the resulting $H^2$ estimate back to $\zeta u$. (Weak divergence-form equations are invariant under $C^2$ boundary charts, $C^2$ flattening preserves uniform ellipticity quantitatively, Tangential $H^2$ estimate near a flat Dirichlet boundary, The normal second derivative is recovered from the equation, Interior $H^2$ regularity for divergence-form equations, Bounded C^k domains and boundary charts)

Gluing: the finite partition lemma assembles the interior and boundary local bounds into the global bound. (A finite partition glues the local interior and boundary $H^2$ estimates)

Global energy bound: testing the zero-trace equation with $u\in H_0^1(\mathrm{\Omega })$ and taking real parts gives $\theta \mathrm{\parallel }Du{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2\le \mathrm{\parallel }f{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}+\sqrt{n}{M}_b\mathrm{\parallel }Du{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}+M_c\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2\mathrm{.}$ Young's inequality absorbs the gradient product and yields $\mathrm{\parallel }Du{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}\le C(\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })})$ with $C=C(n,\theta ,M_b,M_c)$ (Weak Dirichlet solutions for a divergence-form operator, Uniformly elliptic divergence-form operators and their sesquilinear forms, Young's inequality for conjugate real exponents).