*Preprint*

**Inserted:** 14 apr 2009

**Year:** 2009

**Abstract:**

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

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