Let
- $d\in\mathbb N$
- $\Lambda\subseteq\mathbb R^d$ be bounded and open
Let $u\in H_0^1(\Lambda)$ admit a weak Laplacian $\Delta u\in L^2(\Lambda)$. Are we able to show that $$\nabla u\in L^p(\Lambda,\mathbb R^d)\tag1$$ for some $p>2$?
If
- $\partial\Lambda\in C^2$ or
- $\Lambda$ is convex,
then we can even show that $u\in H^2(\Lambda)$ and the desired conclusion is possible (for small $d$) by the Sobolev inequality. However, I wondered whether we can show the weaker result $(1)$ without one of these assumptions. It would be fine for me, if we assume
- $d\le 3$,
- $\Lambda$ is connected or
- $\partial\Lambda$ is Lipschitz