Sharp estimates of the curvature of some free boundaries

Problem's Description

Let $\mu$ be a positive measure with compact support in $\mathbb{R}^n$, $n\geq 3$, and let $K$ be the convex hull of the support of $\mu$. Let $u\geq 0$ be the smallest lower semicontinuous function satisfying $$ u\geq 0, \quad \mu+\Delta u\leq 1, $$ and let $\Omega=\{u>0\}$ be the non-coincidence set for this obstacle problem. The problem is to show that $\Omega$ can be written as a union of balls with centers in $K$. This amounts to saying that $\partial\Omega$ is subject to a natural, and sharp, upper bound for the curvature.

The problem appears in the context of free boundaries for Hele-Shaw flow (or Laplacian growth) and accordingly in the theory of quadrature domains for subharmonic functions and partial balayage. The problem has been solved (positively) in dimension $n=2$, but remains open for $n\geq 3$.

1. Article On the curvature of some free boundaries in higher dimension,

   • Björn Gustafsson,
   • Makoto Sakai

   Anal. Math. Phys. 2 (2012), 247-275., 2012