Q: Does $u \in C^2(\Omega) \cap C(\bar{\Omega})$ imply that $u$ is a solution to Laplace's equation? Does $u \in C^2(\Omega) \cap C(\bar{\Omega})$ imply that $u$ is a solution to Laplace's equation?
I.e. that $\Delta u = 0$ holds.

A: Of course not. All polynomials are in $C^2(\Omega)\cap C(\overline{\Omega})$, and most of them don't satisfy Laplace's equation.
