  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 $\int \left(\nabla u\cdot (A(x)\nabla u) +
\varphi(x) 1_{\{u>0\}}\right) \,\mathrm{d}x \to \text{min}$, where $A(x)$ is an
elliptic matrix with bounded, measurable coefficients and $\varphi$ is not
necessarily locally bounded. We prove universal H\"older 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, $\xi \in \varphi^{-1}(\infty)$. We show that it is determined by the
blow-up rate of $\varphi$ near $\xi$ and we obtain an analytical description of
such cusp geometries.
