Degenerate free discontinuity problems and spectral inequalities in quantitative form

TL;DR

This paper introduces a new geometric-analytic functional with a negative jump set term to analyze free discontinuity problems and derive quantitative spectral inequalities related to Robin boundary eigenvalues.

Contribution

It presents a novel degenerate functional involving obstacle problems and free boundary analysis to establish quantitative inequalities for spectral constants.

Findings

Developed a new functional with negative jump set term

Analyzed the interaction of obstacle problems with free discontinuity

Derived quantitative inequalities for spectral constants in Sobolev-Poincaré inequalities

Abstract

We introduce a new geometric-analytic functional that we analyse in the context of free discontinuity problems. Its main feature is that the geometric term (the length of the jump set) appears with negative sign. This is motivated by searching quantitative inequalities for best constants of Sobolev-Poincar\'e inequalities with trace terms in $\mathbb{R}^n$ which correspond to fundamental eigenvalues associated to semilinear problems for the Laplace operator with Robin boundary conditions. Our method is based on the study of this new, degenerate, functional which involves an obstacle problem in interaction with the jump set. Ultimately, this becomes a mixed free discontinuity/free boundary problem occuring above/at the level of the obstacle, respectively.