Minimization of $\lambda_2(\Omega)$ with a perimeter constraint

Dorin Bucur (1) , Antoine Henrot (2, 3)

1 LAMA - Laboratoire de Mathématiques
2 IECN - Institut Élie Cartan de Nancy
3 CORIDA - Robust control of infinite dimensional systems and applications

We study the problem of minimizing the second Dirichlet eigenvalue for the Laplacian operator among sets of given perimeter. In two dimension, we prove that the optimum exists, is convex, regular, symmetric and its boundary contains exactly two points where the curvature vanishes. In $N$ dimension, we prove existence of a minimizer in a slightly different (relaxed) class and we prove, assuming enough regularity, that this minimizer has cylindrical symmetry.

Mots clés: isoperimetric inequality; eigenvalue; Laplae operator; perimeter constraint

Domaines: Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]; Optimisation et contrôle [math.OC]

Dorin Bucur, Antoine Henrot. Minimization of $\lambda_2(\Omega)$ with a perimeter constraint. 2008. ⟨hal-00201946v1⟩