Inserted: 7 mar 2012
Last Updated: 5 nov 2013
Journal: Arch. Ration. Mech. Anal.
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.