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

Abstract

We investigate the dependence of optimal constants in Poincaré-Sobolev inequalities of planar domains on the region where the Dirichlet condition is imposed. More precisely, we look for the best Dirichlet regions, among closed and connected sets with prescribed total length L (one-dimensional Hausdorff measure), that make these constants as small as possible. We study their limiting behaviour, showing, in particular, that Dirichlet regions homogenize inside the domain with comb-shaped structures, periodically distribuited at different scales and with different orientations. To keep track of these information we rely on a Γ-convergence result in the class of varifolds. This also permits applications to reinforcements of anisotropic elastic membranes. At last, we provide some evidences for a conjecture.

Convex duality for principal frequencies

“…, considered also in [24] and more recently in [2,11,33], among others. The fact that the minimum above is attained in W 1,2 0 (Ω) follows from the compactness of the embedding W 1,2 0 (Ω) ֒→ L q (Ω).…”