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