Abstract
For every \({k \in \mathbb{N}}\), 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 a finite perimeter. The key point is the observation that such quasi-open sets are shape subsolutions for an energy minimizing free boundary problem.
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Communicated by G. Dal Maso
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Bucur, D. Minimization of the k-th eigenvalue of the Dirichlet Laplacian. Arch Rational Mech Anal 206, 1073–1083 (2012). https://doi.org/10.1007/s00205-012-0561-0
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DOI: https://doi.org/10.1007/s00205-012-0561-0