Minimization problems for eigenvalues of the Laplacian

Antoine Henrot (1)

1 CORIDA - Robust control of infinite dimensional systems and applications

This paper is a survey on classical results and open questions about minimization problems concerning the lower eigenvalues of the Laplace operator. After recalling classical isoperimetric inequalities for the two first eigenvalues, we present recent advances on this topic. In particular, we study the minimization of the second eigenvalue among plane convex domains. We also discuss the minimization of the third eigenvalue. We prove existence of a minimizer. For others eigenvalues, we just give some conjectures. We also consider the case of Neumann, Robin and Stekloff boundary conditions together with various functions of the eigenvalues.

Mots clés: Eigenvalues; minimization; isoperimetric inequalities; optimal domain

Domaines: Equations aux dérivées partielles [math.AP]

Antoine Henrot. Minimization problems for eigenvalues of the Laplacian. Journal of Evolution Equations, 2003, 3, pp.443-461. ⟨10.1007/978-3-0348-7924-8_24⟩. ⟨hal-00115548⟩