Global Dirichlet Holder estimate from finite flattened charts

Finite normalized charts. Near each boundary point rotate and translate so that the domain is $x_3>\phi (x^{\mathrm{\prime }})$, with $\phi (0)=0$ and $D\phi (0)=0$. The shear $\mathrm{\Psi }(y^{\mathrm{\prime }},t)=(y^{\mathrm{\prime }},t+\phi (y^{\mathrm{\prime }}))$ maps the actual upper half-patch onto the domain portion of its image, and maps $t=0$ onto the boundary. Its inverse subtracts $\phi$, its Jacobian determinant is one, and $J=D\mathrm{\Psi }$ and $J^{-1}$ are bounded on smaller compact patches. Compactness of the smooth boundary permits a finite initial atlas. Further recenterings at nearby boundary points and rotations to their tangent planes have uniformly bounded derivatives on a common smaller radius: in each initial compact graph patch the normal derivative is bounded away from zero and the graph derivatives are bounded; solving the graph identity and differentiating gives these uniform bounds. Thus arbitrarily small radii $r$ can be used with one fixed bound on the first three chart derivatives, independent of the number of recentered patches.

For $v=u\circ \mathrm{\Psi }$, the chain rule gives $D_{y}v=J^T{D}_{x}u$ and $D_y^{2}v=J^T{D}_x^{2}uJ+{\sum }_k{u}_{x_k}{D}_y^2{\mathrm{\Psi }}_k$. Solving these identities expresses ${\mathrm{\Delta }}_{x}u\circ \mathrm{\Psi }=a^{ij}{D}_{ij}v+b^i{D}_{i}v$, with $a=J^{-1}{J}^{-T}$, $a(0)=I$, and bounded smooth $b$. On a radius-$r$ patch, $\mathrm{\parallel }a-I{\mathrm{\parallel }}_{\mathrm{\infty }}\le Cr$, while $[a{]}_{\alpha }$, $\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{\infty }}$ and $[b{]}_{\alpha }$ are bounded uniformly for $r$ below the fixed atlas radius. Chain/product differences also give two-sided control of the local $C^{2,\alpha }$ norms. In particular the coefficient of the top-order seminorm in this comparison stays bounded as $r\downarrow 0$; terms involving derivatives of charts and derivatives of $u$ of order at most two contribute a fixed-radius multiple of $\mathrm{\parallel }u{\mathrm{\parallel }}_{C^2}$. This follows from $[pq{]}_{\alpha }\le \mathrm{\parallel }p{\mathrm{\parallel }}_{\mathrm{\infty }}[q{]}_{\alpha }+[p{]}_{\alpha }\mathrm{\parallel }q{\mathrm{\parallel }}_{\mathrm{\infty }}$ and $[p\circ \mathrm{\Psi }{]}_{\alpha }\le [p{]}_{\alpha }\mathrm{Lip}(\mathrm{\Psi }{)}^{\alpha }$.

Local freezing. Write $f=\mathrm{\Delta }u$. In a boundary half-patch, ${\mathrm{\Delta }}_{y}v=f\circ \mathrm{\Psi }+(I-a):D^{2}v-b\cdot Dv$ and $v=0$ on the flat face. Apply [F2], with outer radius $r$ and inner radius $r\mathrm{/}2$. The seminorm part of the principal error is bounded by $Cr[D^{2}v{]}_{\alpha }+C_r\mathrm{\parallel }v{\mathrm{\parallel }}_{C^2}$; its supremum contributes only $C_r\mathrm{\parallel }v{\mathrm{\parallel }}_{C^2}$ after rescaling. The drift product and its seminorm contribute only $C_r\mathrm{\parallel }v{\mathrm{\parallel }}_{C^2}$ because $Dv$ is Lipschitz on the convex half-ball. It follows, after the chart comparison in step 2.1, that $[D^{2}u{]}_{\alpha ;\mathrm{i}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{c}\mathrm{h}}\le C_{0}r[D^{2}u{]}_{\alpha ;\mathrm{\Omega }}+C_r(\mathrm{\parallel }f{\mathrm{\parallel }}_{C^{0,\alpha }}+\mathrm{\parallel }u{\mathrm{\parallel }}_{C^2})$, where $C_0$ is independent of small $r$. For an interior ball, extend a cutoff times $f$ by zero, subtract its Newtonian potential, and bound the harmonic remainder by the averaging-kernel argument in [F3]. This gives the same bound without the $C_{0}r$ term. No regularity of a weak solution has been assumed in this use of a priori estimates.

Global seminorm absorption. At a fixed sufficiently small $r$, take boundary centers forming an $r$-net in the compact boundary and interior balls of comparable radii for points farther from the boundary. Use outer patches with fixed larger radii proportional to $r$ and controlled inner patches of radius proportional to $r$; the uniform bi-Lipschitz chart bounds of step 1.1 ensure that each point has a controlled inner patch containing its intersection with a Euclidean ball of radius $cr$, with $c>0$ independent of the number of centers. Compactness selects finitely many. A pair at distance below $cr$ is therefore in one controlled patch; a pair farther apart has Hessian difference quotient at most $2(cr{)}^{-\alpha }\mathrm{\parallel }{D}^{2}u{\mathrm{\parallel }}_{\mathrm{\infty }}$. Taking a maximum of the local estimates, not a sum over patches, yields $[D^{2}u{]}_{\alpha ;\mathrm{\Omega }}\le C_{1}r[D^{2}u{]}_{\alpha ;\mathrm{\Omega }}+C_r(\mathrm{\parallel }f{\mathrm{\parallel }}_{C^{0,\alpha }}+\mathrm{\parallel }u{\mathrm{\parallel }}_{C^2})$. Choose $r$ with $C_{1}r<1\mathrm{/}2$ and absorb. Thus $\mathrm{\parallel }u{\mathrm{\parallel }}_{C^{2,\alpha }}\le C(\mathrm{\parallel }f{\mathrm{\parallel }}_{C^{0,\alpha }}+\mathrm{\parallel }u{\mathrm{\parallel }}_{C^2})$, with the radius now fixed solely by the domain and exponent.

Removing the lower derivative terms. If the stated estimate were false, normalize zero-boundary $C^{2,\alpha }$ functions to obtain $\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{C^{2,\alpha }}=1$ and $\mathrm{\parallel }\mathrm{\Delta }{u}_j{\mathrm{\parallel }}_{C^{0,\alpha }}+\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{\mathrm{\infty }}\to 0$. The finite chart atlas makes the function and gradient fields uniformly equicontinuous from their bounded next derivatives; the Hessians are equicontinuous from their common Hölder bound. Apply [F4] to this finite list of derivative fields. A subsequence converges uniformly on $\bar{\mathrm{\Omega }}$; integration on coordinate segments in interior balls identifies its derivative fields. Since $u_j\to 0$ uniformly, its first and second derivative limits vanish in the interior and, by continuity, on the closure. Thus $\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{C^2}\to 0$. Step 4.1 then gives $1\le C(\mathrm{\parallel }\mathrm{\Delta }{u}_j{\mathrm{\parallel }}_{C^{0,\alpha }}+\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{C^2})\to 0$, a contradiction. This proves the full a priori estimate with only the supremum of $u$ on the right.

All charts, radii and constants were fixed by $\mathrm{\Omega }$ and $\alpha$ before the solution was chosen. Reflection and the Newtonian kernel enter through [F2], flattening and coefficient perturbation were derived in steps 1.1--3.1, and the finite-cover and compactness arguments in steps 4.1--5.1 complete the claimed global estimate. No published boundary or global Schauder theorem is an input.