Global Schauder regularity for the weak Dirichlet Laplacian

Reduction to zero boundary values. Put $F:=f+\mathrm{\Delta }g$ on $\bar{\mathrm{\Omega }}$. Since $g\in C^{2,\alpha }(\bar{\mathrm{\Omega }})$ and $\mathrm{\Delta }g\in C^{0,\alpha }(\bar{\mathrm{\Omega }})$, one has $F\in C^{0,\alpha }(\bar{\mathrm{\Omega }})$ with $\mathrm{\parallel }F{\mathrm{\parallel }}_{C^{0,\alpha }}\le \mathrm{\parallel }f{\mathrm{\parallel }}_{C^{0,\alpha }}+C(\mathrm{\Omega })\mathrm{\parallel }g{\mathrm{\parallel }}_{C^{2,\alpha }}$. If $u$ is the weak solution of the problem of [F1], then $w:=u-g$ lies in $H_0^1(\mathrm{\Omega })$ and, for every $\phi \in H_0^1(\mathrm{\Omega })$, $\int \mathrm{\nabla }w\cdot \mathrm{\nabla }\phi =\int \mathrm{\nabla }u\cdot \mathrm{\nabla }\phi -\int \mathrm{\nabla }g\cdot \mathrm{\nabla }\phi =\int f\phi +\int \mathrm{\Delta }g\phi =\int F\phi$, the middle identity for $g$ being the weak form of $-\mathrm{\Delta }g$ for $C^2$ functions (approximate $\phi$ by $C_c^{\mathrm{\infty }}$ functions and integrate by parts). So $w$ is a weak solution of the zero-boundary problem $-\mathrm{\Delta }w=F$; conversely, if such a $w$ is shown to be $C^{2,\alpha }$ up to the boundary with $w=0$ on $\mathrm{\partial }\mathrm{\Omega }$, then $u=w+g$ is the required solution. It suffices to prove the zero-boundary statement for $F\in C^{0,\alpha }(\bar{\mathrm{\Omega }})$: find $w\in H_0^1(\mathrm{\Omega })$ with $-\mathrm{\Delta }w=F$ weakly, $w\in C^{2,\alpha }(\bar{\mathrm{\Omega }})$ and $\mathrm{\parallel }w{\mathrm{\parallel }}_{C^{2,\alpha }}\le C\mathrm{\parallel }F{\mathrm{\parallel }}_{C^{0,\alpha }}$.

Extending and mollifying the data. By [F2] extend $F$ to $F^{\mathrm{♯}}\in C_c^{0,\alpha }({\mathbb{R}}^n)$ with $\mathrm{\parallel }{F}^{\mathrm{♯}}{\mathrm{\parallel }}_{C^{0,\alpha }}\le C_1\mathrm{\parallel }F{\mathrm{\parallel }}_{C^{0,\alpha }(\bar{\mathrm{\Omega }})}$, and put $F_{\epsilon }:=F^{\mathrm{♯}}\ast {\rho }_{\epsilon }$ as in [F3]; then $M:={\sup }_{\epsilon }\mathrm{\parallel }{F}_{\epsilon }{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }{F}^{\mathrm{♯}}{\mathrm{\parallel }}_{\mathrm{\infty }}\le C_1\mathrm{\parallel }F{\mathrm{\parallel }}_{C^{0,\alpha }(\bar{\mathrm{\Omega }})}$ and ${\sup }_{\epsilon }[F_{\epsilon }{]}_{0,\alpha }\le C_1\mathrm{\parallel }F{\mathrm{\parallel }}_{C^{0,\alpha }(\bar{\mathrm{\Omega }})}$, while $F_{\epsilon }\to F$ uniformly on $\bar{\mathrm{\Omega }}$.

The approximating weak solutions and the initial energy bound. Choose $p>n$, for instance $p=n+1$. For each $\epsilon$, $F_{\epsilon }{\mathrm{\mid }}_{\mathrm{\Omega }}\in L^p(\mathrm{\Omega })\subset H^{-1}(\mathrm{\Omega })$, so [F1] gives a unique weak solution $w_{\epsilon }\in H_0^1(\mathrm{\Omega })$ of $-\mathrm{\Delta }{w}_{\epsilon }=F_{\epsilon }$. Testing the weak equation with $w_{\epsilon }$ (or its complex conjugate) and using Poincar'e gives $\mathrm{\parallel }{w}_{\epsilon }{\mathrm{\parallel }}_{H^1}\le C\mathrm{\parallel }{F}_{\epsilon }{\mathrm{\parallel }}_{L^2}\le C\mathrm{\parallel }F{\mathrm{\parallel }}_{C^{0,\alpha }}$, uniformly in $\epsilon$.

Uniform $W^{2,p}$ bound, including the $L^p$ term. Use the finite-exponent bootstrap in the proof of [F4], not just its final a priori estimate. For $n>2$ set $q_0=\min \{p,2n\mathrm{/}(n-2)\}$; for $n=2$ set $q_0=p$. The energy estimate of step 3.1 and the first-order Sobolev embedding control $\mathrm{\parallel }{w}_{\epsilon }{\mathrm{\parallel }}_{L^{q_0}}$. The supplier proof chooses a fixed shift $\lambda$ and a finite list $q_0\le q_1<\cdots <q_m=p$ (with no further step when $q_0=p$), where $q_{i+1}=\min \{p,n{q}_i\mathrm{/}(n-2q_i)\}$ while $2q_i<n$, and $q_{i+1}=p$ once $2q_i\ge n$. At the first exponent, the shifted strong-solvability estimate for $(\lambda -\mathrm{\Delta })z=F_{\epsilon }+\lambda w_{\epsilon }$, together with energy uniqueness, identifies $z=w_{\epsilon }$ and bounds $\mathrm{\parallel }{w}_{\epsilon }{\mathrm{\parallel }}_{W^{2,q_0}}$ by $C(\mathrm{\parallel }{F}_{\epsilon }{\mathrm{\parallel }}_{L^{q_0}}+\mathrm{\parallel }{w}_{\epsilon }{\mathrm{\parallel }}_{L^{q_0}})$. At each later exponent $q_{i+1}$, the embedding in [F5] bounds $\mathrm{\parallel }{w}_{\epsilon }{\mathrm{\parallel }}_{L^{q_{i+1}}}$ by the preceding $W^{2,q_i}$ norm; the next shifted estimate and energy uniqueness then give the $W^{2,q_{i+1}}$ bound. Every $\mathrm{\parallel }{F}_{\epsilon }{\mathrm{\parallel }}_{L^{q_i}}$ is bounded by $C\mathrm{\parallel }F{\mathrm{\parallel }}_{C^{0,\alpha }}$, and the list is finite, so induction gives $\mathrm{\parallel }{w}_{\epsilon }{\mathrm{\parallel }}_{W^{2,p}}\le C(\mathrm{\parallel }{F}_{\epsilon }{\mathrm{\parallel }}_{L^p}+\mathrm{\parallel }{w}_{\epsilon }{\mathrm{\parallel }}_{H^1})\le C\mathrm{\parallel }F{\mathrm{\parallel }}_{C^{0,\alpha }}$, uniformly in $\epsilon$. This controls the $L^p(w_{\epsilon })$ term left explicit in the supplier's final a priori estimate. Applying [F5] with $k=2$, $p>n$, each $w_{\epsilon }$ has a $C^{1,\gamma }(\bar{\mathrm{\Omega }})$ representative, for any fixed $0<\gamma <1-n\mathrm{/}p$, with uniformly bounded norm. Its trace is zero because $w_{\epsilon }\in H_0^1(\mathrm{\Omega })$; [F1] identifies this trace with the boundary values of the continuous representative.

Interior smoothness. Fix a point $x_0\in \mathrm{\Omega }$ and a ball $B⋐\mathrm{\Omega }$ around it. The function $F_{\epsilon }$ is smooth near $\bar{B}$; choose $\theta \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ with $\theta =1$ on a neighbourhood of $\bar{B}$ and set $H_{\epsilon }:=\theta F_{\epsilon }$, a compactly supported smooth function. By [F6], $N{H}_{\epsilon }\in C^{\mathrm{\infty }}({\mathbb{R}}^n)$ and $-\mathrm{\Delta }N{H}_{\epsilon }=H_{\epsilon }=F_{\epsilon }$ on $B$. Hence $\mathrm{\Delta }(w_{\epsilon }-N{H}_{\epsilon })=0$ weakly on $B$: for every $\phi \in C_c^{\mathrm{\infty }}(B)$, $\int \mathrm{\nabla }(w_{\epsilon }-N{H}_{\epsilon })\cdot \mathrm{\nabla }\phi =\int F_{\epsilon }\phi -\int F_{\epsilon }\phi =0$. By the local smoothness of weakly harmonic functions in [F6], $w_{\epsilon }-N{H}_{\epsilon }$ agrees on $B$ with a smooth function; since $N{H}_{\epsilon }$ is smooth, $w_{\epsilon }$ agrees on $B$ with a smooth function. As $B$ and $x_0$ were arbitrary, $w_{\epsilon }\in C^{\mathrm{\infty }}(\mathrm{\Omega })$, and it satisfies $-\mathrm{\Delta }{w}_{\epsilon }=F_{\epsilon }$ pointwise in $\mathrm{\Omega }$.

A uniform supremum bound with the weak maximum principle's sign. Choose $R>0$ and $x_0$ with $\mathrm{\Omega }\subseteq B_R(x_0)$ and put $q(x):=M(R^2-\mathrm{\mid }x-x_0{\mathrm{\mid }}^2)\mathrm{/}(2n)$, where $M:={\sup }_{\epsilon }\mathrm{\parallel }{F}_{\epsilon }{\mathrm{\parallel }}_{\mathrm{\infty }}$. Then $q\ge 0$ on $\bar{\mathrm{\Omega }}$ and $\mathrm{\Delta }q=-M$. For a real-valued solution component $v_{\epsilon }$ with datum $F_{\epsilon }$ satisfying $\mathrm{\mid }{F}_{\epsilon }\mathrm{\mid }\le M$, one has $\mathrm{\Delta }(v_{\epsilon }-q)=M-F_{\epsilon }\ge 0,\mathrm{\Delta }(-v_{\epsilon }-q)=M+F_{\epsilon }\ge 0.$ Both comparison functions are in $C^2(\mathrm{\Omega })\cap C(\bar{\mathrm{\Omega }})$ by steps 4.1 and 4.2, and their boundary values are $-q\le 0$. The weak maximum principle [F7] therefore gives $v_{\epsilon }-q\le 0$ and $-v_{\epsilon }-q\le 0$, hence $\mathrm{\mid }{v}_{\epsilon }\mathrm{\mid }\le q\le M{R}^2\mathrm{/}(2n)$. If the data are complex, apply this argument to the real and imaginary parts separately; then $\mathrm{\mid }{w}_{\epsilon }\mathrm{\mid }\le \sqrt{2}M{R}^2\mathrm{/}(2n)$. By step 2.1, $M\le C\mathrm{\parallel }F{\mathrm{\parallel }}_{C^{0,\alpha }(\bar{\mathrm{\Omega }})}$, so this is a uniform $C^0$ bound.

Uniform $C^{2,\alpha }$ bounds and the limit. The functions $w_{\epsilon }$ lie in $C^2(\mathrm{\Omega })\cap C^0(\bar{\mathrm{\Omega }})$ by steps 4.1 and 4.2, vanish on $\mathrm{\partial }\mathrm{\Omega }$, and satisfy $-\mathrm{\Delta }{w}_{\epsilon }=F_{\epsilon }$ pointwise with $F_{\epsilon }\in C^{0,\alpha }(\bar{\mathrm{\Omega }})$ and $\mathrm{\parallel }{F}_{\epsilon }{\mathrm{\parallel }}_{C^{0,\alpha }}\le C_1\mathrm{\parallel }F{\mathrm{\parallel }}_{C^{0,\alpha }}$. Apply [F8] to each real component of $w_{\epsilon }$ with right-hand side the corresponding component of $-F_{\epsilon }$ (the estimate is unchanged by this sign), and combine the component bounds if the data are complex. Using step 5.1, $\mathrm{\parallel }{w}_{\epsilon }{\mathrm{\parallel }}_{C^{2,\alpha }(\bar{\mathrm{\Omega }})}\le C_7(\mathrm{\parallel }{w}_{\epsilon }{\mathrm{\parallel }}_{C^0(\mathrm{\Omega })}+\mathrm{\parallel }{F}_{\epsilon }{\mathrm{\parallel }}_{C^{0,\alpha }(\bar{\mathrm{\Omega }})})\le C_8\mathrm{\parallel }F{\mathrm{\parallel }}_{C^{0,\alpha }(\bar{\mathrm{\Omega }})},$ uniformly in $\epsilon$. The boundary Schauder estimate gives uniform $C^{2,\alpha }$ control in each member of a finite cover of $\bar{\mathrm{\Omega }}$ by interior balls and flattened boundary half-boxes. Thus the function, gradient and Hessian components are equicontinuous and pointwise bounded on $\bar{\mathrm{\Omega }}$; if the functions are complex, list their real and imaginary components separately. By [F9], a subsequence of this finite-dimensional vector-valued family converges uniformly to limits $(w,G,H)$. On every ball $B⋐\mathrm{\Omega }$, the fundamental theorem of calculus along segments in $B$ and uniform convergence give $Dw=G$ and $DG=H$. The limits $G,H$ are continuous on $\bar{\mathrm{\Omega }}$, so these derivatives extend continuously to the boundary; pointwise convergence of the Hessian difference quotients gives $[D^{2}w{]}_{0,\alpha ;\mathrm{\Omega }}\le {\mathrm{lim\hspace{0.17em}inf}}_{\epsilon }[D^2{w}_{\epsilon }{]}_{0,\alpha ;\mathrm{\Omega }}$. Hence $w\in C^{2,\alpha }(\bar{\mathrm{\Omega }})$ with the stated bound. Uniform convergence of $F_{\epsilon }$ and of the second derivatives gives $-\mathrm{\Delta }w=F$ pointwise in $\mathrm{\Omega }$ and $w=0$ on $\mathrm{\partial }\mathrm{\Omega }$.

Identification with the weak solution. The limit $w\in C^2(\bar{\mathrm{\Omega }})$ with $w=0$ on $\mathrm{\partial }\mathrm{\Omega }$ satisfies $\int \mathrm{\nabla }w\cdot \mathrm{\nabla }\phi =\int F\phi$ for every $\phi \in H_0^1(\mathrm{\Omega })$: for $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ this is integration by parts, and $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ is dense in $H_0^1(\mathrm{\Omega })$. By [F1] the weak solution of the zero-boundary problem is unique, so $w$ is the unique weak solution $w=u-g$ of the original problem as identified in step 1.1. Therefore $u=w+g\in C^{2,\alpha }(\bar{\mathrm{\Omega }})$ solves $-\mathrm{\Delta }u=f$ pointwise and $u=g$ on $\mathrm{\partial }\mathrm{\Omega }$, and $\mathrm{\parallel }u{\mathrm{\parallel }}_{C^{2,\alpha }}\le \mathrm{\parallel }w{\mathrm{\parallel }}_{C^{2,\alpha }}+\mathrm{\parallel }g{\mathrm{\parallel }}_{C^{2,\alpha }}\le C\mathrm{\parallel }F{\mathrm{\parallel }}_{C^{0,\alpha }}+C\mathrm{\parallel }g{\mathrm{\parallel }}_{C^{2,\alpha }}\le C^{\mathrm{\prime }}(\mathrm{\parallel }f{\mathrm{\parallel }}_{C^{0,\alpha }}+\mathrm{\parallel }g{\mathrm{\parallel }}_{C^{2,\alpha }})$ by step 1.1. This is the displayed estimate of the statement.

Conclusion. The weak Dirichlet problem has exactly one solution by [F1], and steps 2.1-7.1 show that this solution is the limit of the smooth approximating solutions, is of class $C^{2,\alpha }(\bar{\mathrm{\Omega }})$ with the stated bound, and solves the equation classically. In particular the Laplace operator with Dirichlet boundary values on a bounded $C^{2,\alpha }$ domain has the Schauder a priori estimate on weak solutions, a fact used as the base point of the method of continuity. The proof uses only the fixed exponent $p>n$ in the weak-to-strong step, and the finiteness of all constants is uniform in the mollification parameter.