It is well-known, see also this thread, that if $\Omega\subset \mathbb{R}^n$ is a bounded domain with smooth boundary, then the eigenfunctions of the Laplacian (with zero-Dirichlet boundary conditions) are real analytic in $\Omega$ and smooth up to the boundary.
1) I was wondering if there exists an explicit example, say on a domain in $\mathbb{R}^2$, of an eigenfunction, which fails to be real analytic up to the boundary?
2) What if we additionally require the boundary to be real analytic (instead of only smooth)? The argument I am familiar with makes use of Sobolev embeddings to conclude smoothness up to the boundary. Since there is no 'real analytic Sobolev embedding' I would guess that even if the boundary is real analytic, the eigenfunction in general will not be real analytic up to the boundary. Is there anyone who can confirm my conjecture?
PS: By real analytic up to the boundary I mean that every point $x\in \partial\Omega$ has an open neighbourhood $x\in U_x\subseteq \mathbb{R}^n$ such that there exists a real analytic function $\tilde{f}:U\rightarrow \mathbb{R}$ whose restriction $\tilde{f}|_{U\cap \overline{\Omega}}$ coincides with the eigenfunction $f$ on $U\cap \overline{\Omega}$.
Kind regards Dennis