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

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

1 LAMA - Laboratoire de Mathématiques
2 Dipartimento di Matematica [Pisa]
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 dimensions, we prove that the optimum exists, is convex, regular, and its boundary contains exactly two points where the curvature vanishes. In $N$ dimensions, we prove a more general existence theorem for a class of functionals which is decreasing with respect to set inclusion and $\gamma$ lower semicontinuous.

Mots clés: Dirichlet Laplacian; eigenvalues; perimeter constraint; isoperimetric problem

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

Dorin Bucur, Giuseppe Buttazzo, Antoine Henrot. Minimization of $\lambda_2(\Omega)$ with a perimeter constraint. Indiana University Mathematics Journal, 2009, 58 (6), pp.2709-2728. ⟨hal-00201946v3⟩