Mixed Dirichlet-Neumann eigenvalue problem

Question

Let $\Omega\subset\Bbb R^2$ be a bounded $C^2$ domain. Let $\partial\Omega=\partial\Omega_1\cup\partial\Omega_2$. Does anyone know about the existence of eigenvalues and eigenfunctions for the following mixed eigenvalue problem: $\left\{\begin{array}{l l}-\Delta u=\lambda u&\quad x\in\Omega\\ u=0&\quad x\in\partial\Omega_1\\ \frac{\partial u}{\partial n}=0&\quad x\in\partial\Omega_2 \end{array}\right.$