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 ∫ ∇ u · ( A ( x ) ∇ u ) + φ ( x ) 1 { u > 0 } d x → 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 ∈ φ - 1 ( ∞ ) . 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.