Minimization of $λ_2(Ω)$ with a perimeter constraint

Details Minimization of $λ_2(Ω)$ with a perimeter constraint Minimization of $λ_2(Ω)$ with a perimeter constraint

2009-04-14

arxiv.org/abs/0904.2193v2

Indiana University Mathematics Journal 58, 6 (2009) 2709-2728

Mathematics

Analysis of PDEs

Indiana University Mathematics Journal (2009) to appear

Scientific paper

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.

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