On the Lipschitz regularity and asymptotic behaviour of the free boundary for classes of minima of inhomogeneous two-phase Alt–Caffarelli functionals in Orlicz spaces

Abstract

In this paper, we study classes of minimizers of inhomogeneous two-phase Alt–Caffarelli functionals of the type $$\begin{aligned} {\mathcal {J}}_G(u,\Omega ) = \int _{\Omega } \Big [ G({\vert \nabla u \vert }) + f_1(x)H_1(u^+) + f_2(x) H_2(u^-) + Q(h_1, h_2)(u)(x) \Big ]\mathrm{d}x, \end{aligned}$$ on a bounded domain $$\Omega \subset {\mathbb {R}}^n$$ , where $$G, H_1$$ and $$H_2$$ are power-like N-functions, $$f_1, f_2 \in L^q(\Omega )$$ for suitable $$n \le q \le \infty $$ , and $$h_1, h_2 \in L^{\infty }(\Omega )$$ . Holder and non-degeneracy estimates for minima are obtained and in the particular case where such minimizers are weak solutions of non-singular PDEs we provide log-Lipschitz type estimates. In the sequel, since the Alt–Caffarelli–Friedman monotonicity formula is missing in our context and there is a mistake in a proof of Lipschitz continuity for minimizers in Zheng et al. (Monatsh Math 172:441–475, 2013), we extend the results of Braga et al. (Ann Inst H Poincare Anal Non Lineaire 31(4):823–850, 2014) establishing the Lipschitz regularity for more general class of minima under the additional condition of small Lebesgue density on one of the phases along the free boundary. We finish this paper with a result that establishes density estimates from below for the positive and negative phase on points inside the contact set between the free boundaries in the case where minimizers are not Lipschitz. Such estimates allow us to provide a preliminary full description of the free boundary for any minima even if the Lipschitz regularity (as optimal regularity) is unknown.