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Degenerate free discontinuity problems and spectral inequalities in quantitative form. (English) Zbl 1472.49064

Authors’ 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 a negative sign. This is motivated by searching quantitative inequalities for the best constants of Sobolev-Poincaré 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.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
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