# Minimization of the k-th eigenvalue of the Dirichlet Laplacian

created by bucur on 07 Mar 2012
modified on 05 Nov 2013

[BibTeX]

Published Paper

Inserted: 7 mar 2012
Last Updated: 5 nov 2013

Journal: Arch. Ration. Mech. Anal.
Volume: 206
Pages: 1073-1083
Year: 2012

Abstract:

For every $k\in N$ we prove the existence of a quasi-open set minimizing the k-th eigenvalue of the Dirichlet Laplacian among all sets of prescribed Lebesgue measure. Moreover, we prove that every minimizer is bounded and has finite perimeter. The key point is the observation that such quasi-open sets are shape subsolutions for an energy minimizing free boundary problem.