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Minimizing the Second Eigenvalue of the Laplace Operator with Dirichlet Boundary Conditions

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Abstract

In this paper, we are interested in the minimization of the second eigenvalue of the Laplacian with Dirichlet boundary conditions amongst convex plane domains with given area. The natural candidate to be the optimum was the ``stadium'', a convex hull of two identical tangent disks. We refute this conjecture. Nevertheless, we prove the existence of a minimizer. We also study some qualitative properties of the minimizer (regularity, geometric properties).

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Authors and Affiliations

  1. Ecole des Mines and Institut Elie Cartan Nancy, B.P. 239 54506, Vandoeuvre les Nancy Cedex, France

    Antoine Henrot

  2. Institut de Recherche Mathématique Avancée, Université Louis Pasteur, 67084, Strasbourg Cedex, France

    Edouard Oudet

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  1. Antoine Henrot
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  2. Edouard Oudet
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Henrot, A., Oudet, E. Minimizing the Second Eigenvalue of the Laplace Operator with Dirichlet Boundary Conditions. Arch. Rational Mech. Anal. 169, 73–87 (2003). https://doi.org/10.1007/s00205-003-0259-4

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