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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: $\left\{\begin{array}{l l}-\Delta u=\lambda u&\quad x\in\Omega\\ u=0&\quad x\in\partial\Omega_1\\ \frac{\partial u}{\partial n}=0&\quad x\in\partial\Omega_2 \end{array}\right.$

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  • $\begingroup$ related: Eigenvalues of the 1D laplacian with mixed boundary conditions $\endgroup$ – Surb Aug 17 '15 at 10:07
  • $\begingroup$ @Surb, I think the 1D version is quite different, do you know anything about the $\Bbb R^2$ or $\Bbb R^n$ case? $\endgroup$ – Ellya Aug 17 '15 at 10:18
  • $\begingroup$ To be honest I forgot almost everything I knew about PDE... However I believe that searching a bit on google scholar you should find a few paper dealing with this problem. $\endgroup$ – Surb Aug 18 '15 at 6:33
  • $\begingroup$ I remember that met such problems in the literature while I was looking for the eigenvalues of the $p$-Laplace operator. $\endgroup$ – Surb Aug 18 '15 at 6:46
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    $\begingroup$ Searching for "Zaremba eigenproblem" would be a start: e.g., arxiv.org/pdf/1411.0071.pdf $\endgroup$ – user147263 Aug 19 '15 at 3:17

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