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

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

1 GFM - Grupo de Física Matemática - Group of Mathematical Physics
2 CORIDA - Robust control of infinite dimensional systems and applications

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.

Mots clés: Dirichlet Laplacian; Neumann Laplacian; Shape optimization; Eigenvalues

Domaines: Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]

Pedro R.S. Antunes, Antoine Henrot. On the range of the first two Dirichlet and Neumann eigenvalues of the Laplacian. Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences, 2011, 467 (2130), pp.1577-1603. ⟨10.1098/rspa.2010.0444⟩. ⟨hal-00511096⟩