I am interested in the Laplace operator $\Delta$ on a concave polygon. When the polygon is convex, it is known that $\Delta: H^2(\Omega) \rightarrow L^2(\Omega)$ is boundedly invertible. In addition, when there is an obtuse angle a similar result holds for $\Delta: H_w^2(\Omega)\rightarrow L_w^2(\Omega)$, where $H_w^2, L_w^2$ are weighted Sobolev spaces. In this case, the weight function depends on the distance from the corners, $L^2$ is continuously embedded in $L^2_w$, and the Laplacian is again isomorphism. For a special choice of weights (depending on the size of the angles in the polygon), $A^{-1}: L_w^2(\Omega)\rightarrow L_w^2(\Omega)$ is a compact operator. Now, I wonder is it possible to have an orthonormal basis for $L_w^2(\Omega)$ of eigenfunctions for $\Delta$ when $\Omega$ is a concave polygon?
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$\begingroup$ It should go exactly same as the usual case by spectral theory of compact operators. You can also manually do it by the minimax approach, like, e.g., in Jost's PDE book. $\endgroup$– timurCommented May 13, 2011 at 3:03
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I believe the answer should be yes, since even concave polygons satisfy an "exterior cone condition". See Elliptic partial differential equations of second order by Gilbarg and Trudinger for details. If course, this might be more high-tech than necessary.
Of course, I am assuming here that you understand your operator as the operator with Dirichlet boundary conditions...
