Abstract

In this paper we look for the domains minimizing the $h$th eigenvalue of the Dirichlet-Laplacian $\lambda_h$ with a constraint on the diameter. Existence of an optimal domain is easily obtained and is attained at a constant width body. In the case of a simple eigenvalue, we provide nonstandard (i.e., nonlocal) optimality conditions. Then we address the question of whether the disk is an optimal domain in the plane, and we give the precise list of the 17 eigenvalues for which the disk is a local minimum. We conclude by some numerical simulations showing the 20 first optimal domains in the plane.

Keywords

  1. Dirichlet eigenvalues
  2. spectral geometry
  3. diameter constraint
  4. body of constant width

MSC codes

  1. 35P15
  2. 49R05
  3. 35J25
  4. 49Q10

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Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 5337 - 5361
ISSN (online): 1095-7154

History

Submitted: 20 December 2017
Accepted: 6 August 2018
Published online: 2 October 2018

Keywords

  1. Dirichlet eigenvalues
  2. spectral geometry
  3. diameter constraint
  4. body of constant width

MSC codes

  1. 35P15
  2. 49R05
  3. 35J25
  4. 49Q10

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