Let $\Omega\subset\mathbb R^d$ be a compact and connected subset with smooth (or piecewise smooth) boundary denoted by $\partial \Omega$. Let $\Gamma^+, \Gamma^- \subset\partial \Omega$ be such that
$$\Gamma^+\cap \Gamma^-=\emptyset \quad\mbox{ and }\quad \Gamma^+\cup \Gamma^- =\partial \Omega.$$
Consider the collection of eigenvalues $\lambda_k$ and eigenvectors $u_k\neq 0$ defined as follows:
\begin{eqnarray} &&-\Delta u_k ~=~ \lambda_k u_k \quad\mbox{on}\quad \Omega \\ && u|_{\Gamma^+} ~=~ 0 ~=~ \frac{\partial u_k}{\partial n }\Big|_{\Gamma^-} \end{eqnarray}
I look for references on the regularity and the closed-form expression (e.g. $d=2$) of $u_k$? Many thanks.
