Calculus of Variations and Geometric Measure Theory

D. Zucco

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

created by zucco on 16 Oct 2018
modified on 30 Mar 2019


Accepted Paper

Inserted: 16 oct 2018
Last Updated: 30 mar 2019

Journal: Calc. Var. Partial Differential Equations
Year: 2019

ArXiv: 1810.06384 PDF


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.