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In [David-Toro 15] and [David-Engelstein-Toro 19], (some of) the authorsstudied almost minimizers for functionals of the type first studied by Alt andCaffarelli in [Alt-Caffarelli 81] and Alt, Caffarelli and Friedman in[Alt-Caffarelli-Friedman 84]. In this paper we study the regularity of almostminimizers 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 coefficientsetting.

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