Minimization of $\lambda_{2}(\Omega)$ with a perimeter constraint
Minimization of $\lambda_{2}(\Omega)$ with a perimeter constraint
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 …