On the range of the first two Dirichlet and Neumann eigenvalues of the Laplacian

Pedro R.S. Antunes; Antoine Henrot

hal-00511096, version 1

On the range of the first two Dirichlet and Neumann eigenvalues of the Laplacian

Pedro R.S. Antunes ¹, Antoine Henrot () ^2, 3

Proceedings of the Royal Society of London Series A Containing Papers of a Mathematical and Physical Character 467, 2130 (2011) 1577-1603

Abstract: In this paper we study the set of points, in the plane, defined by $\{(x,y)=(\lambda_1(\Omega),\lambda_2(\Omega)),\ |\Omega|=1\},$ where $(\lambda_1(\Omega),\lambda_2(\Omega))$ are either the two first eigenvalues of the Dirichlet-Laplacian, or the two first non trivial eigenvalues of the Neumann-Laplacian. We consider the case of general open sets together with the case of convex open domains. We give some qualitative properties of these sets, show some pictures obtained through numerical computations and state several open problems.

1: Grupo de Física Matemática - Group of Mathematical Physics (GFM)
Universidade de Lisboa
2: Institut Elie Cartan Nancy (IECN)
CNRS : UMR7502 – INRIA – Université Henri Poincaré - Nancy I – Université Nancy II – Institut National Polytechnique de Lorraine (INPL)
3: CORIDA (INRIA Nancy - Grand Est / IECN / LMAM)
INRIA – CNRS : UMR7502 – Université de Lorraine

hal-00511096, version 1
http://hal.archives-ouvertes.fr/hal-00511096
oai:hal.archives-ouvertes.fr:hal-00511096
From: Antoine Henrot <>
Submitted on: Monday, 23 August 2010 17:32:30
Updated on: Monday, 2 May 2011 10:31:47

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