Regularity of solution for Poisson equation on bounded domains

Question

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain and consider $-\Delta u=f$ in $\Omega$, $u=\varphi$ on $\partial\Omega$.

For balls with some radius the following result is known: Let $\varphi \in C^{2, \alpha}(\bar{B}), f \in C^\alpha(\bar{B})$. Then the Dirichlet problem, $\Delta u=f$ in $B, u=\varphi$ on $\partial B$, is uniquely solvable for a function $u \in C^{2, \alpha}(\bar{B})$.

Adding some regularity on $\Omega$ we can probably get the following: Assume that $\varphi\in C^{2,\alpha}(\overline{\Omega})$ and $f\in C^{\alpha}(\overline{\Omega})$. Then there exists a solution $u\in C^{2,\alpha}(\overline{\Omega})$.

Question I suspect that $\partial\Omega\in C^2$ is sufficient. Is there even less needed? If anybody knows a reference I would be grateful.

Answer 1

To have unique solvability for the Dirichlet problem you can assume a lot less regularity on the boundary, but then you won't necessarily have that $u$ is $C^{2,\alpha}$ up to the boundary. For example, in Elliptic Partial Differential Equations of Second Order by Gilbarg and Trudinger, we have:

Theorem 6.13: Let $L$ be strictly elliptic in a bounded domain $\Omega$ with $c\leqslant 0$, and let $f$ and the coefficients of $L$ be bounded and belong to $C^\alpha(\Omega)$. Suppose that $\Omega$ satisfies an exterior sphere condition at every boundary point. Then, if $\varphi$ is continuous on $\partial \Omega$, the Dirichlet problem, $$Lu=f \text{ in } \Omega, \qquad u=\varphi \text{ on } \partial \Omega$$ has a (unique) solution $u\in C^0(\overline \Omega)\cap C^{2,\alpha}(\Omega)$.

In fact, they claim that the assumption that $\Omega$ satisfies an exterior sphere condition at every boundary point can be relaxed to only satisfying the exterior cone condition (they leave it has an exercise to prove this). Also, in the following theorem (Theorem 6.14), they prove what you suspected, that is, if $\Omega$ is $C^{2,\alpha}$ and $\varphi \in C^{2,\alpha}(\overline\Omega)$ then there is a unique solution $u\in C^{2,\alpha}(\overline\Omega)$.