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

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.

Résumé

Dans [5, 6] les auteurs ont étudié les fonctions presque minimales pour des fonctionnelles comme celles d’Alt et Caffarelli [1] , et d’Alt, Caffarelli et Friedman [2]. Dans ce papier on étudie la régularité des fonctions presque minimales pour des fonctionnelles d’énergie à coefficients variables (contrairement à [1, 2, 5,6] qui se placent dans le cadre du Laplacian). On prouve que ces fonctions sont Lipschitziennes juqu’à la frontière, et à travers, généralisant ainsi les résultats de [6] au cas de coefficients variables.