Calculus of Variations and Geometric Measure Theory

D. Bucur - G. Buttazzo - A. Henrot

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

created by buttazzo on 14 Apr 2009



Inserted: 14 apr 2009

Year: 2009


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.

Keywords: eigenvalues, isoperimetric problem, Dirichlet Laplacian, perimeter constraint