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.

在David等人中。（Adv Math 350：1109–1192，2019）和David and Toro（具有自由边界的几乎最小化器的正则性。微分和PDE的微积分，2020），作者研究了几乎最小化器用于Alt和Caffarelli（ J Reine Angew Math 325：105-144，1981）和Alt等。（Trans Am Math Soc 282：431–461，1984年）。在本文中，我们研究了几乎最小化器对具有可变系数的能量泛函的正则性（与Alt和Caffarelli，J Reine Angew Math 325：105-144，1981； Alt等人，Trans Am Math Soc 282：431-461 ，1984年； David等人，Adv Math 350：1109–1192，2019； David和Toro，带有自由边界的几乎最小化器的正则性（微积分和PDE的微积分，2020），仅涉及“拉普拉斯”设置）。我们证明Lipschitz直到自由边界的规则性，