Minimization of the k-th eigenvalue of the Dirichlet Laplacian

By Dorin Bucur. Archive for Rational Mechanics and Analysis, 2012

120 citations recorded by the source corpus.

For every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${k \in \mathbb{N}}$$\end{document}, 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c0835b8c239859b7e4a151be474b33d761aabdfb

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. S. Terracini
2. Bozhidar Velichkov
3. Dario Mazzoleni
4. Bernard Helffer
5. Gianmaria Verzini

In Computational Theory and Mathematics

1. S. Terracini
2. Bozhidar Velichkov
3. Dario Mazzoleni
4. Bernard Helffer

In Mathematical Physics

1. S. Terracini
2. Bozhidar Velichkov
3. Dario Mazzoleni
4. Bernard Helffer

Citations in the corpus

• Regularity of the optimal sets for some spectral functionals (2016, background): “(ii) (Bucur [6]; Mazzoleni-Pratelli [33]) There is a solution to the shape optimization problem min { λ1(Ω) + · · ·+ λk(Ω) : Ω ⊂ R quasi-open, |Ω| = 1 } .”