On the Bernoulli problem with unbounded jumps

Abstract

We investigate Bernoulli free boundary problems prescribing infinite jump conditions. The mathematical set-up leads to the analysis of non-differentiable minimization problems of the form $\int \left(\nabla u\cdot (A(x)\nabla u) + φ(x) 1_{\{u>0\}}\right) \,\mathrm{d}x \to \text{min}$, where $A(x)$ is an elliptic matrix with bounded, measurable coefficients and $φ$ is not necessarily locally bounded. We prove universal Hölder continuity of minimizers for the one- and two-phase problems. Sharp regularity estimates along the free boundary are also obtained. Furthermore, we perform a thorough analysis of the geometry of the free boundary around a point $ξ$ of infinite jump, $ξ\in φ^{-1}(\infty)$. We show that it is determined by the blow-up rate of $φ$ near $ξ$ and we obtain an analytical description of such cusp geometries.