Optimizing the first Dirichlet eigenvalue of the Laplacian with an obstacle

Abstract

Inside a fixed bounded domain Ω of the plane, we look for the best compact connected set K, of given perimeter, in order to maximize the first Dirichlet eigenvalue λ 1 (Ω \ K). We discuss some of the qualitative properties of the maximizers, moving toward existence, regularity and geometry. Then we study the problem in specific domains: disks, rings, and, more generally, disks with convex holes. In these situations, we prove symmetry and in some cases non symmetry results, identifying the solution.We choose to work with the outer Minkowski content as the "good" notion of perimeter. Therefore, we are led to prove some new properties for it as its lower semicontinuity with respect to the Hausdorff convergence and the fact that the outer Minkowski content is equal to the Hausdorff lower semicontinuous envelope of the classical perimeter.

Dirichlet conditions in Poincaré–Sobolev inequalities: the sub-homogeneous case

“…In the homogeneous case p = q the explicit formula given in Remark 3.4 for Poincaré-Sobolev constants with mixed boundary conditions is still true; this implies the validity of Theorem 3.6. Consequently, with only minor changes from the linear case p = q = 2, previously treated in [27,29] (see also [17,28] for other related results), one can prove the following result. Theorem 6.1 (Γ-convergence in the homogeneous case).…”

Spectral partitions for Sturm–Liouville problems

“…Therefore, according to whether one wishes to optimize among partitions or points, problem (3) is a matter of optimal partition or of optimal location (see [5,6,12] for some optimal partition problems with cost functionals depending on the eigenvalues of the Laplacian in higher dimension and [2,4,13,15,16] for some optimal location problems of non-spectral cost functionals). Problem (3) can also be seen as a one-dimensional version of the problems introduced in [11,17,18], where the issue was how to best place a Dirichlet boundary condition in a two-dimensional membrane in order to optimize the first eigenvalue of an elliptic operator. Some possible physical interpretations of problem (3) are as follows.…”

On the quantitative isoperimetric inequality in the plane

“…Remark 2.6. We also recall that P DG (K) ≤ P (K) if K ⊂ R 2 is a compact connected set, as remarked in [19].…”

“…We We will also use the following semicontinuity result, analogous to the Golab Theorem for the Minkowski perimeter in the plane, proved by Henrot and Zucco in [19]: Theorem 2.3. Let {K n } ⊂ R 2 be a sequence contained in K(Ω) converging to a set K ∈ K(Ω) in the Hausdorff metric.…”