I stumbled over this question in the context of PDE theory and thought that maybe somebody here knows whether the following is true or not?
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 $u \in C^1(\overline{U})$?
If you feel that any information is missing, please let me know.
