An Overview of Almost Minimizers of Bernoulli-Type Functionals

In David and Toro (Calc Var 54(1):455–524, 2015) and David et al. (Adv Math 350:1109–1192, 2019), the authors studied almost minimizers for functionals of the type first studied by Alt and Caffarelli (J Reine Angew Math325:105–144, 1981) and Alt, Caffarelli and Friedman (Trans Am Math Soc 282:431–461, 1984). In this chapter, we study the regularity of almost minimizers to energy functionals with variable coefficients (as opposed to Alt and Caffarelli (J Reine Angew Math325:105–144, 1981), Alt, Caffarelli and Friedman (Trans Am Math Soc 282:431–461, 1984), David et al. (Adv Math 350:1109–1192, 2019), and David and Toro (Calc Var 54(1):455–524, 2015) that 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 (Calc Var 54(1):455–524, 2015) to the variable coefficient setting.