Boundary Schauder estimate for the Dirichlet problem

Reduction to zero boundary values. Put $w:=u-g$ on $\mathrm{\Omega }$. Since $g\in C^{2,\alpha }(\bar{\mathrm{\Omega }})\subseteq C^2(\mathrm{\Omega })\cap C^0(\bar{\mathrm{\Omega }})$, the function $w$ lies in $C^2(\mathrm{\Omega })\cap C^0(\bar{\mathrm{\Omega }})$, vanishes on $\mathrm{\partial }\mathrm{\Omega }$, and satisfies $Lw=F$ pointwise in $\mathrm{\Omega }$ with $F:=f-Lg$. The product inequality $[ab{]}_{\alpha }\le \mathrm{\parallel }a{\mathrm{\parallel }}_{\mathrm{\infty }}[b{]}_{\alpha }+[a{]}_{\alpha }\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{\infty }}$ gives $[Lg{]}_{0,\alpha ;\mathrm{\Omega }}\le C_n(\mathrm{\Lambda }[D^{2}g{]}_{0,\alpha ;\mathrm{\Omega }}+K\mathrm{\parallel }{D}^{2}g{\mathrm{\parallel }}_{\mathrm{\infty };\mathrm{\Omega }}+\mathrm{\parallel }b{\mathrm{\parallel }}_{\mathrm{\infty }}[Dg{]}_{0,\alpha ;\mathrm{\Omega }}+[b{]}_{0,\alpha ;\mathrm{\Omega }}\mathrm{\parallel }Dg{\mathrm{\parallel }}_{\mathrm{\infty };\mathrm{\Omega }}+\mathrm{\parallel }c{\mathrm{\parallel }}_{\mathrm{\infty }}[g{]}_{0,\alpha ;\mathrm{\Omega }}+[c{]}_{0,\alpha ;\mathrm{\Omega }}\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty };\mathrm{\Omega }})$ and ${\sup }_{\mathrm{\Omega }}\mathrm{\mid }Lg\mathrm{\mid }\le C_n(\mathrm{\Lambda }+M)\mathrm{\parallel }g{\mathrm{\parallel }}_{C^{2,\alpha }(\mathrm{\Omega })}$; in each flattened half-box the lower-field seminorms are bounded by the derivative suprema using integration along segments, and a finite cover plus the separation bound controls pairs in different charts; hence $F\in C^{0,\alpha }(\bar{\mathrm{\Omega }})$ with $\mathrm{\parallel }F{\mathrm{\parallel }}_{C^{0,\alpha }(\bar{\mathrm{\Omega }})}\le \mathrm{\parallel }f{\mathrm{\parallel }}_{C^{0,\alpha }(\bar{\mathrm{\Omega }})}+C(n,\alpha ,\mathrm{\Lambda },K,M,\mathrm{\Omega })\mathrm{\parallel }g{\mathrm{\parallel }}_{C^{2,\alpha }(\bar{\mathrm{\Omega }})}$, and ${\sup }_{\mathrm{\Omega }}\mathrm{\mid }w\mathrm{\mid }\le \mathrm{\parallel }u{\mathrm{\parallel }}_{C^0(\mathrm{\Omega })}+\mathrm{\parallel }g{\mathrm{\parallel }}_{C^{2,\alpha }(\bar{\mathrm{\Omega }})}$. It thus suffices to show that every such zero-boundary $w$ with $Lw=F\in C^{0,\alpha }(\bar{\mathrm{\Omega }})$ satisfies $w\in C^{2,\alpha }(\bar{\mathrm{\Omega }})$ and $\mathrm{\parallel }w{\mathrm{\parallel }}_{C^{2,\alpha }(\mathrm{\Omega })}\le C(\mathrm{\parallel }w{\mathrm{\parallel }}_{\mathrm{\infty };\mathrm{\Omega }}+\mathrm{\parallel }F{\mathrm{\parallel }}_{C^{0,\alpha }(\bar{\mathrm{\Omega }})})$ with $C=C(n,\alpha ,\lambda ,\mathrm{\Lambda },K,M,\mathrm{\Omega })$; adding $g$ back gives the statement.

A finite relative cover with a Lebesgue number. By [F3], for every $x\in \mathrm{\partial }\mathrm{\Omega }$ there is a flattened chart ${\mathrm{\Psi }}_x:Q_{r_x}\to U_x$ with ${\mathrm{\Psi }}_x(Q_{r_x}^{+})=U_x\cap \mathrm{\Omega }$ and ${\mathrm{\Psi }}_x(Q_{r_x}^{\mathrm{\prime }}\times \{0\})=U_x\cap \mathrm{\partial }\mathrm{\Omega }$. Put ${\widehat{V}}_x:={\mathrm{\Psi }}_x(Q_{r_x\mathrm{/}2})$, an ambient open neighborhood of $x$ that includes points on both sides of the flattened boundary. These sets cover $\mathrm{\partial }\mathrm{\Omega }$; compactness gives finitely many, indexed by $\mathrm{\ell }=1,\dots ,N$, and their union $V$ is an open neighborhood of $\mathrm{\partial }\mathrm{\Omega }$ in ${\mathbb{R}}^n$. Thus some $\delta >0$ satisfies $\{y\in \mathrm{\Omega }:\mathrm{dist}(y,\mathrm{\partial }\mathrm{\Omega })<\delta \}\subseteq V$. The set ${\mathrm{\Omega }}_{\delta }:=\{y\in \mathrm{\Omega }:\mathrm{dist}(y,\mathrm{\partial }\mathrm{\Omega })\ge \delta \}$ is compact in $\mathrm{\Omega }$. With $R:=\delta \mathrm{/}4$, its balls $B_R(z)$ have doubled balls inside $\mathrm{\Omega }$, and compactness yields finitely many centres $z_1,\dots ,z_P$ whose inner balls cover ${\mathrm{\Omega }}_{\delta }$. The finite family consisting of the ambient open sets ${\widehat{V}}_{\mathrm{\ell }}$ and the balls $B_R(z_j)$ covers $\bar{\mathrm{\Omega }}$ relative to $\bar{\mathrm{\Omega }}$. Let ${\delta }_0>0$ be a Lebesgue number for this relative open cover, so every pair $x,y\in \bar{\mathrm{\Omega }}$ with $\mathrm{\mid }x-y\mathrm{\mid }<{\delta }_0$ lies in one common member.

Estimates on the boundary members. Fix $\mathrm{\ell }\in \{1,\dots ,N\}$, write $\mathrm{\Psi }={\mathrm{\Psi }}_{\mathrm{\ell }}$, $r=r_{\mathrm{\ell }}$ and let $\tilde{w}:=w\circ \mathrm{\Psi }$ on $Q_r^{+}$, with pullback operator $\tilde{L}$ as in [F3]. Then $\tilde{w}\in C^2(Q_r^{+})\cap C^0(\overline{Q_r^{+}})$ because $\mathrm{\Psi }$ is a $C^{2,\alpha }$ diffeomorphism of a neighbourhood of $\overline{Q_r^{+}}$ onto a neighbourhood of $\overline{U_{\mathrm{\ell }}\cap \mathrm{\Omega }}$ and $w\in C^2(\mathrm{\Omega })\cap C^0(\bar{\mathrm{\Omega }})$; moreover $\tilde{w}=0$ on the flat face, since that face maps onto $U_{\mathrm{\ell }}\cap \mathrm{\partial }\mathrm{\Omega }$ where $w=0$. The pullback satisfies $\tilde{L}\tilde{w}=\tilde{h}$ pointwise in $Q_r^{+}$ with $\tilde{h}:=F\circ \mathrm{\Psi }$, which lies in $C^{0,\alpha }(Q_r^{+})\cap C^0(\overline{Q_r^{+}})$ by composition, and $\mathrm{\parallel }\tilde{h}{\mathrm{\parallel }}_{0,\alpha ;Q_r^{+}}^{\ast }\le C_{\mathrm{\ell }}\mathrm{\parallel }F{\mathrm{\parallel }}_{C^{0,\alpha }(\bar{\mathrm{\Omega }})}$ with $C_{\mathrm{\ell }}$ depending on the chart. Applying the quoted boundary estimate [F4] to $\tilde{w}$ gives $\mathrm{\parallel }\tilde{w}{\mathrm{\parallel }}_{2,\alpha ;Q_{r\mathrm{/}2}^{+}}^{\ast }\le C_{\mathrm{\ell }}^{\mathrm{\prime }}(\mathrm{\parallel }w{\mathrm{\parallel }}_{\mathrm{\infty };\mathrm{\Omega }}+\mathrm{\parallel }F{\mathrm{\parallel }}_{C^{0,\alpha }(\bar{\mathrm{\Omega }})}),$ with $C_{\mathrm{\ell }}^{\mathrm{\prime }}$ depending on $n,\alpha$, the ellipticity constants and coefficient bounds of $\tilde{L}$ (hence on the chart and on $\lambda ,\mathrm{\Lambda },K,M$). By the norm equivalence of [F3], the last display controls $\sup \mathrm{\mid }{D}^{\beta }w\mathrm{\mid }$ for $\mathrm{\mid }\beta \mathrm{\mid }\le 2$ and $[D^{2}w{]}_{0,\alpha }$ over $\mathrm{\Psi }(Q_{r\mathrm{/}2}^{+})=U_{\mathrm{\ell }}\cap V_{\mathrm{\ell }}$ with $V_{\mathrm{\ell }}:={\mathrm{\Psi }}_{\mathrm{\ell }}(Q_{r_{\mathrm{\ell }}\mathrm{/}2}^{+})$; finitely many charts give one constant $C_2$.

Estimates on the interior members. Fix $j\in \{1,\dots ,P\}$ and let $B_{2R}(z_j)\subseteq \mathrm{\Omega }$. On $B_{2R}(z_j)$ the function $w$ is of class $C^2$ and bounded, and $Lw=F$ pointwise with $F\in C^{0,\alpha }(B_{2R}(z_j))$; the coefficient bounds $[A{]}_{0,\alpha }\le K$ and $\mathrm{\parallel }b{\mathrm{\parallel }}_{C^{0,\alpha }}+\mathrm{\parallel }c{\mathrm{\parallel }}_{C^{0,\alpha }}\le M$ hold there. By [F2] with radius $2R$, $\mathrm{\parallel }w{\mathrm{\parallel }}_{2,\alpha ;B_R(z_j)}^{\ast }\le C_j(\mathrm{\parallel }w{\mathrm{\parallel }}_{\mathrm{\infty };\mathrm{\Omega }}+\mathrm{\parallel }F{\mathrm{\parallel }}_{\mathrm{\infty };\mathrm{\Omega }}+[F{]}_{0,\alpha ;\mathrm{\Omega }}),$ where $C_j$ depends on $n,\alpha ,\lambda ,\mathrm{\Lambda }$ and the dimensionless bounds formed with the radius $2R$; as there are finitely many $j$ and all radii are comparable to $\delta$, the numbers $C_j$ are bounded by a constant $C_1$ depending on $n,\alpha ,\lambda ,\mathrm{\Lambda },K,M$ and on the cover. In particular ${\sup }_{B_R(z_j)}\mathrm{\mid }{D}^{\beta }w\mathrm{\mid }\le R^{-\mathrm{\mid }\beta \mathrm{\mid }}\mathrm{\parallel }w{\mathrm{\parallel }}_{2,\alpha ;B_R(z_j)}^{\ast }$ for $\mathrm{\mid }\beta \mathrm{\mid }\le 2$ and $[D^{2}w{]}_{0,\alpha ;B_R(z_j)}\le R^{-2-\alpha }\mathrm{\parallel }w{\mathrm{\parallel }}_{2,\alpha ;B_R(z_j)}^{\ast }$.

Summation and conclusion. The interior balls $B_R(z_j)$ together with the boundary-chart sets ${\widehat{V}}_{\mathrm{\ell }}\cap \bar{\mathrm{\Omega }}$ cover $\bar{\mathrm{\Omega }}$; on each boundary-chart set, its portion in $\bar{\mathrm{\Omega }}$ lies in ${\mathrm{\Psi }}_{\mathrm{\ell }}(\stackrel{‾}{Q_{r_{\mathrm{\ell }}\mathrm{/}2}^{+}})$, where step 1.3 supplies the estimate. Thus for every $\mathrm{\mid }\beta \mathrm{\mid }\le 2$, ${\sup }_{\mathrm{\Omega }}\mathrm{\mid }{D}^{\beta }w\mathrm{\mid }\le {\sum }_j{\sup }_{B_R(z_j)}\mathrm{\mid }{D}^{\beta }w\mathrm{\mid }+{\sum }_{\mathrm{\ell }}{\sup }_{{\widehat{V}}_{\mathrm{\ell }}\cap \mathrm{\Omega }}\mathrm{\mid }{D}^{\beta }w\mathrm{\mid }\le C_3(\mathrm{\parallel }w{\mathrm{\parallel }}_{\mathrm{\infty };\mathrm{\Omega }}+\mathrm{\parallel }F{\mathrm{\parallel }}_{C^{0,\alpha }(\bar{\mathrm{\Omega }})})$ by steps 2.1 and 1.3. For the H"older seminorm let $x,y\in \mathrm{\Omega }$, $x\mathrm{\ne }y$. If $\mathrm{\mid }x-y\mathrm{\mid }<{\delta }_0$, step 1.2 gives a member of the relative cover containing both points, and steps 2.1 or 1.3 bound the corresponding difference quotient; if $\mathrm{\mid }x-y\mathrm{\mid }\ge {\delta }_0$, then $\mathrm{\mid }{D}^{2}w(x)-D^{2}w(y)\mathrm{\mid }\mathrm{/}\mathrm{\mid }x-y{\mathrm{\mid }}^{\alpha }\le 2{\delta }_0^{-\alpha }{\sup }_{\mathrm{\Omega }}\mathrm{\mid }{D}^{2}w\mathrm{\mid }$, already controlled. Therefore $\mathrm{\parallel }w{\mathrm{\parallel }}_{C^{2,\alpha }(\mathrm{\Omega })}={\sum }_{j=0}^2{\sup }_{\mathrm{\Omega }}{\max }_{\mathrm{\mid }\beta \mathrm{\mid }=j}\mathrm{\mid }{D}^{\beta }w\mathrm{\mid }+[D^{2}w{]}_{0,\alpha ;\mathrm{\Omega }}\le C(\mathrm{\parallel }w{\mathrm{\parallel }}_{\mathrm{\infty };\mathrm{\Omega }}+\mathrm{\parallel }F{\mathrm{\parallel }}_{C^{0,\alpha }(\bar{\mathrm{\Omega }})})$ with $C=C(n,\alpha ,\lambda ,\mathrm{\Lambda },K,M,\mathrm{\Omega })$. Since the boundary-chart sets are ambient neighborhoods of each boundary point and the controlled coordinate functions are $C^{2,\alpha }$ up to the flat face, the derivatives $D^{\beta }w$ for $\mathrm{\mid }\beta \mathrm{\mid }\le 2$ extend continuously to $\bar{\mathrm{\Omega }}$. Hence $w\in C^{2,\alpha }(\bar{\mathrm{\Omega }})$ with the same norm over $\mathrm{\Omega }$, as required by [F1].

The estimate for $u$. By step 1.1 and step 3.1, $\mathrm{\parallel }u{\mathrm{\parallel }}_{C^{2,\alpha }(\bar{\mathrm{\Omega }})}\le \mathrm{\parallel }w{\mathrm{\parallel }}_{C^{2,\alpha }(\mathrm{\Omega })}+\mathrm{\parallel }g{\mathrm{\parallel }}_{C^{2,\alpha }(\bar{\mathrm{\Omega }})}\le C(\mathrm{\parallel }u{\mathrm{\parallel }}_{C^0(\mathrm{\Omega })}+\mathrm{\parallel }f{\mathrm{\parallel }}_{C^{0,\alpha }(\bar{\mathrm{\Omega }})}+\mathrm{\parallel }g{\mathrm{\parallel }}_{C^{2,\alpha }(\bar{\mathrm{\Omega }})})$ after enlarging $C$ to absorb the constants of step 1.1; the constant depends on $n,\alpha ,\lambda ,\mathrm{\Lambda },K,M$ and on the finite cover, i.e. on $\mathrm{\Omega }$ only through its $C^{2,\alpha }$ charts and radii. The strict range $0<\alpha <1$ enters through the quoted boundary estimate [F4] and the interior estimate [F2]; the Dirichlet condition enters through $w=0$ on the boundary, which is exactly the zero-data hypothesis of [F4] on each flat face. No solvability, compactness of the operator or boundary regularity beyond the $C^{2,\alpha }$ chart hypothesis is used, and it is the Dirichlet condition and the $C^{2,\alpha }$ boundary that make the estimate possible at every boundary point.