Abstract

In this paper we give a method to geometrically modify an open set such that the first $k$ eigenvalues of the Dirichlet Laplacian and its perimeter are not increasing, its measure remains constant, and both perimeter and diameter decrease below a certain threshold. The key point of the analysis relies on the properties of the shape subsolutions for the torsion energy. As well, we apply this result to prove existence of solutions for shape optimization problems of spectral type with both measure and perimeter constraints.

Keywords

  1. shape optimization
  2. eigenvalues
  3. Dirichlet Laplacian

MSC codes

  1. 49R05
  2. 49J35
  3. 35P15

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Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 4451 - 4466
ISSN (online): 1095-7154

History

Submitted: 21 October 2014
Accepted: 16 September 2015
Published online: 24 November 2015

Keywords

  1. shape optimization
  2. eigenvalues
  3. Dirichlet Laplacian

MSC codes

  1. 49R05
  2. 49J35
  3. 35P15

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