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Degenerate Free Discontinuity Problems and Spectral Inequalities in Quantitative Form

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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.

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Acknowledgements

The first and the third authors were supported by the LabEx PERSYVAL-Lab GeoSpec (ANR-11-LABX-0025-01) and ANR SHAPO (ANR-18-CE40-0013).

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Authors and Affiliations

  1. CNRS, LAMA, Univ. Savoie Mont Blanc, 73000, Chambéry, France

    Dorin Bucur & Mickaël Nahon

  2. DICATAM, Sezione di Matematica, Università degli Studi di Brescia, Via Branze 43, 25133, Brescia, Italy

    Alessandro Giacomini

Authors
  1. Dorin Bucur
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  2. Alessandro Giacomini
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  3. Mickaël Nahon
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Correspondence to Dorin Bucur.

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Communicated by G. Dal Maso.

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Bucur, D., Giacomini, A. & Nahon, M. Degenerate Free Discontinuity Problems and Spectral Inequalities in Quantitative Form. Arch Rational Mech Anal 242, 453–483 (2021). https://doi.org/10.1007/s00205-021-01688-7

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  • DOI: https://doi.org/10.1007/s00205-021-01688-7

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