Estimate on the $C^{1,\alpha}(\bar \Omega)$ norm of solution of linear elliptic Neumann problem

Question

Let $\Omega$ be a bounded smooth domain with $$-\Delta u + ku = f$$ $$\partial_\nu u|_{\partial\Omega} = 0$$ where $k > 0$ is a constant and $f \in L^\infty(\Omega)$. It follows that $u \in H^2(\Omega)\cap C^{1,\alpha}(\bar \Omega)$.

I want an estimate on $\lVert u \rVert_{C^{1,\alpha}(\bar \Omega)}$ (or at least an estimate on the $L^\infty$ norm of the gradient of $u$) in terms of the data but I cannot find any literature for the Neumann problem giving an a priori estimate.

Does anyone know this estimate? I'd rather keep dimension as general as possible so I don't want to use Sobolev embeddings to obtain this.

Answer 1

Really a comment, but too long.

@Math604 is right, the key is to use the $L^p$ theory (sometimes referred to as strong solutions). Indeed, the right-hand side being in $L^\infty$ is a borderline case for the regularity theory (as always s£^% hits the fan when $p=\infty$ is involved): One may naively hope from the usual elliptic regularity motto "$Lu\in L^p\Rightarrow u\in W^{2,p}$" that $u\in W^{2,\infty}$ if the right-hand side $f\in L^\infty$, but this is known to fail in this bordeline case. Instead, since $L^\infty\subset L^p$ for all $p<\infty$ (!) you can get $u\in W^{2,p}$ for all $p<\infty$. And then use Sobolev embedding to conclude that $u\in C^{1,\alpha}$ (up to the boundary) for all $\alpha<1$ (the embedding constant must blow-up as $p\to+\infty$). This is well referenced for the Dirichlet boundary values, much less indeed for the Neumann case (as usual, one of us should definitely write the ultimate textbook on the Neumann problem one of these days!).

So I would start looking-up for the $L^p$ regularity theory for the Neumann problem and then follow this roadmap.