Minimization of $\lambda_{2}(\Omega)$ with a perimeter constraint Dorin BucurGiuseppe ButtazzoAntoine Henrot 49Q1049J4549R5035P1547A75Dirichlet Laplacianeigenvaluesperimeter constraintisoperimetric problem 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. Indiana University Mathematics Journal 2009 text pdf 10.1512/iumj.2009.58.3768 10.1512/iumj.2009.58.3768 en Indiana Univ. Math. J. 58 (2009) 2709 - 2728 state-of-the-art mathematics http://iumj.org/access/