Consider that $u\in H^1(\Omega)$ with $\Delta u\in L^2(\Omega)$ (in the distributional sense) such that for some $\lambda>0$ we have that:

$$\begin{cases} \Delta u(x)=\lambda u(x), & x\in\Omega\\ \dfrac{\partial u}{\partial\nu} (x)=0, & x\in\partial\Omega\end{cases}$$

We assume $\Omega\subseteq\mathbb{R}^2$ to be an open, connected, bounded and has a uniform Lipschitz boundary.

**1)** Is it true that $u\in H^2(\Omega)$? If this is not true:

**2)** Is it true that $u\in C(\overline{\Omega})$? If this is not true:

**3)** How can we prove that $u\in L^{\infty}(\Omega)$?

I know that (3) is valid from the inequality posted here: Contractivity of Neumann Laplacean

But I do not know how to prove that inequality. Maybe it can be done in an easier way...

**I wonder if there is an estimate of the form:**

$\Vert u\Vert_{\infty}\leq c\lambda^{\alpha}$, **where $c$ is a constant depending on $\Omega$?**

I know that such estimates hold for for the Dirichlet laplacian. I found some references about the problem here:

but they do not represent counter-examples for any of my questions.

P.S. I found that (2) might also be true from that post: Is the linear span of the Neumann eigenfunctions dense in $C(\overline{D})$ but I did not understand the argument. Is there any clear way of getting more information about the regularity of the eigenfunction $u$ of $\lambda$?

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