Regularity for almost-minimizers of variable coefficient Bernoulli-type functionals

Guy David, Max Engelstein, Mariana Smit Vega Garcia, Tatiana Toro

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1 Scopus citations

Abstract

In David et al. (Adv Math 350:1109–1192, 2019) and David and Toro (Regularity of almost minimizers with free boundary. Calculus of variations and PDEs, 2020), the authors studied almost minimizers for functionals of the type first studied by Alt and Caffarelli (J Reine Angew Math 325:105–144, 1981) and Alt et al. (Trans Am Math Soc 282:431–461, 1984). In this paper we study the regularity of almost minimizers to energy functionals with variable coefficients (as opposed to Alt and Caffarelli, J Reine Angew Math 325:105–144, 1981; Alt et al., Trans Am Math Soc 282:431–461, 1984; David et al., Adv Math 350:1109–1192, 2019; David and Toro, Regularity of almost minimizers with free boundary. Calculus of variations and PDEs, 2020) which deal only with the “Laplacian” setting). We prove Lipschitz regularity up to, and across, the free boundary, fully generalizing the results of David and Toro (Regularity of almost minimizers with free boundary. Calculus of variations and PDEs, 2020) to the variable coefficient setting.

Original languageEnglish (US)
Pages (from-to)2131-2169
Number of pages39
JournalMathematische Zeitschrift
Volume299
Issue number3-4
DOIs
StatePublished - Dec 2021

Bibliographical note

Funding Information:
GD was partially supported by the ANR, programme blanc GEOMETRYA, ANR-12-BS01-0014, the European H2020 Grant GHAIA 777822, and the Simons Collaborations in MPS Grant 601941, GD. ME was partially supported by NSF DMS 2000288, NSF DMS 1703306 and by David Jerison’s grant NSF DMS 1500771. TT was partially supported by NSF grant DMS-1664867, by the Craig McKibben& Sarah Merner Professorship in Mathematics, and by the Simons Foundation Fellowship 614610.

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

Keywords

  • Almost-minimizer
  • Alt–Caffarelli functional
  • Alt–Caffarelli–Friedman
  • Free boundary problem

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