Regularity of a free boundary with application to the Pompeiu problem

Luis A. Caffarelli, Lavi Karp, Henrik Shahgholian

In the unit ball B(0,1), let $u$ and $\Omega$ (a domain in $\mathbb{R}$) solve the following overdetermined problem: $$Δu =χ_Ω\quad \hbox{in} B(0,1), \qquad 0 \in \partial Ω, \qquad u=|\nabla u…