orthonormal basis of eigenvectors for laplacian on a concave polygon

2011-03-24T16:46:19Z

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?

Answer by Helge for orthonormal basis of eigenvectors for laplacian on a concave polygon

2011-03-25T00:04:04Z

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...

orthonormal basis of eigenvectors for laplacian on a concave polygon - MathOverflow

orthonormal basis of eigenvectors for laplacian on a concave polygon

Answer by Helge for orthonormal basis of eigenvectors for laplacian on a concave polygon