On the Bernoulli problem with unbounded jumps

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 $\int (\mathrm{\nabla }u\cdot (A(x)\mathrm{\nabla }u)+\phi (x)1_{\{u>0\}})\mathrm{d}x\to \text{min}$, where $A(x)$ is an elliptic matrix with bounded, measurable coefficients and $\phi$ 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 $\xi$ of infinite jump, $\xi \in {\phi }^{-1}(\mathrm{\infty })$. We show that it is determined by the blow-up rate of $\phi$ near $\xi$ and we obtain an analytical description of such cusp geometries.