Regularity theorem for PDE: $-\Delta u \in C(\overline{\Omega})$ implies $u\in C^1(\overline{\Omega})$?

Question

I stumbled over this question in the context of PDE theory: Let $U$ be connected,open and bounded in $\mathbb{R}^n$ and $u \in C^0(\overline{U}) \cap C^2(U)$ and $\Delta u \in C^0(\overline{U})$ with $u|_{\partial U} = \Delta u|_{\partial U} = 0.$ Does this imply that $u \in C^1(\overline{U})$?

Answer 1

If $U$ is a bounded domain with regular boundary then, one approach is the following: Consider the problem

$$ \left\{ \begin{array}{rl} -\Delta u=f &\mbox{ in}\ U, \\ u=0 &\mbox{on } \partial U. \end{array} \right. $$

Once $f\in C(\overline{U})$, we also have that $f\in L^p(U)$ for each $p\in [1,\infty)$. This implies in particular (see 1 chapter 9) that $u\in W_0^{2,p}(U)$ for each $p\in [1,\infty)$.

Now take $p$ big enough to conclude that $W_0^{2,p}(U)$ is continuously embedded in $C^1(\overline{U})$.