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

TL;DR

This paper extends the regularity results for almost-minimizers of Bernoulli-type functionals to variable coefficient settings, proving Lipschitz continuity up to the free boundary, thus generalizing previous Laplacian-based results.

Contribution

It establishes Lipschitz regularity for almost-minimizers of variable coefficient Bernoulli functionals, broadening the scope from constant coefficient cases.

Findings

Lipschitz regularity up to the free boundary

Generalization from Laplacian to variable coefficients

Regularity results applicable to a broader class of functionals

Abstract

In [David-Toro 15] and [David-Engelstein-Toro 19], (some of) the authors studied almost minimizers for functionals of the type first studied by Alt and Caffarelli in [Alt-Caffarelli 81] and Alt, Caffarelli and Friedman in [Alt-Caffarelli-Friedman 84]. In this paper we study the regularity of almost minimizers to energy functionals with variable coefficients (as opposed to [DT15, DET19. AC 81] and [ACF84] which deal only with the "Laplacian" setting). We prove Lipschitz regularity up to, and across, the free boundary, generalizing the results of [David-Toro 15] to the variable coefficient setting.