Dirichlet conditions in Poincar\'e-Sobolev inequalities: the sub-homogeneous case

TL;DR

This paper studies how the placement of Dirichlet boundary conditions affects the optimal constants in Poincaré-Sobolev inequalities on planar domains, revealing homogenization phenomena and applications to elastic membranes.

Contribution

It introduces a variational approach using $mbda$-convergence to analyze optimal Dirichlet regions with prescribed length, uncovering homogenization and structural behaviors.

Findings

Dirichlet regions homogenize into comb-shaped structures

Optimal regions depend on the total length constraint

Applications to anisotropic elastic membrane reinforcement

Abstract

We investigate the dependence of optimal constants in Poincar\'e- 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 Dirichler 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 $\Gamma$-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.