Best regularity for elliptic PDE with Neumann data

Question

Consider $-\Delta u + \lambda u = f$ on $\Omega$ with BC $\partial_\nu u = 0$ on $\partial\Omega$ where $\Omega$ is a bounded smooth domain and $\lambda > 0$.

If $f \in L^\infty(\Omega)$, what's the best regularity I can get for the solution $u$?

I think I can get $u \in H^2(\Omega) \cap C^{0,\alpha}(\bar \Omega)$ for some exponent $\alpha < 1.$

Can I do better than this?

Answer 1

You have the inclusion $f \in L^\infty(\Omega) \subset L^p(\Omega)$ for all $1 \le p \le \infty$ due to the fact that $\Omega$ is bounded. Then the $L^p-$based elliptic regularity for the Neumann problem will give you that $u \in W^{2,p}(\Omega)$ for all $1 < p < \infty$. Using this and the Sobolev embeddings will then bump you up to $u \in C^{1,\alpha}(\bar{\Omega})$ for each $0 < \alpha < 1$. This is the best Holder regularity you can get with $f \in L^\infty$.