Multiphase free discontinuity problems: Monotonicity formula and regularity results

  • Alessandro Giacomini

    DICATAM, Sezione di Matematica, Università degli Studi di Brescia, Via Branze 43, 25133 Brescia, Italy
  • Dorin Bucur

    Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France
  • Ilaria Fragalà

    Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy
Multiphase free discontinuity problems: Monotonicity formula and regularity results cover

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Abstract

The purpose of this paper is to analyze regularity properties of local solutions to free discontinuity problems characterized by the presence of multiple phases. The key feature of the problem is related to the way in which two neighboring phases interact: the contact is penalized at jump points, while no cost is assigned to no-jump interfaces which may occur at the zero level of the corresponding state functions. Our main results state that the phases are open and the jump set (globally considered for all the phases) is essentially closed and Ahlfors regular. The proof relies on a multiphase monotonicity formula and on a sharp collective Sobolev extension result for functions with disjoint supports on a sphere, which may be of independent interest.

Cite this article

Alessandro Giacomini, Dorin Bucur, Ilaria Fragalà, Multiphase free discontinuity problems: Monotonicity formula and regularity results. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 5, pp. 1553–1582

DOI 10.1016/J.ANIHPC.2020.12.003