On the Bernoulli problem with unbounded jumps

TL;DR

This paper studies Bernoulli free boundary problems with infinite jump conditions, establishing regularity of minimizers and analyzing the free boundary's geometric structure near points with infinite jumps.

Contribution

It introduces a new analysis of free boundary problems with unbounded jump conditions, providing regularity results and geometric descriptions of cusps at infinite jump points.

Findings

Proved universal Hölder continuity of minimizers.

Obtained sharp regularity estimates at the free boundary.

Characterized free boundary geometry near infinite jump points.

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 $\int \left(\nabla u\cdot (A(x)\nabla u) + \varphi(x) 1_{\{u>0\}}\right) \,\mathrm{d}x \to \text{min}$, 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.