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Inside a fixed bounded domain $\Omega$ of the plane, we look for the bestcompact connected set $K$, of given perimeter, in order to maximize the firstDirichlet eigenvalue $\lambda_1(\Omega\setminus K)$. We discuss some of thequalitative properties of the maximizers, moving toward existence, regularityand geometry. Then we study the problem in specific domains: disks, rings, and,more generally, disks with convex holes. In these situations, we prove symmetryand, in some cases non symmetry results, identifying the solution.  We choose to work with the outer Minkowski content as the "good" notion ofperimeter. Therefore, we are led to prove some new properties for it as itslower semicontinuity with respect to the Hausdorff convergence and the factthat the outer Minkowski content is equal to the Hausdorff lower semicontinuousenvelope of the classical perimeter.

Optimizing the first Dirichlet eigenvalue of the Laplacian with an obstacle