Calculus of Variations and Geometric Measure Theory

D. Bucur - A. Giacomini - M. Nahon

Degenerate free discontinuity problems and spectral inequalities in quantitative form

created by bucur on 27 Oct 2020


Submitted Paper

Inserted: 27 oct 2020

Year: 2020

ArXiv: 2010.05883 PDF


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-Poincaré inequalities with trace terms in ℝn 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 discontinuityfree boundary problem occuring aboveat the level of the obstacle, respectively.