Mathematics – Analysis of PDEs
Scientific paper
2000-10-02
Ann. of Math. (2) 151 (2000), no. 1, 269--292
Mathematics
Analysis of PDEs
24 pages
Scientific paper
In the unit ball B(0,1), let $u$ and $\Omega$ (a domain in $\R$) solve the following overdetermined problem: $$\Delta u =\chi_\Omega\quad \hbox{in} B(0,1), \qquad 0 \in \partial \Omega, \qquad u=|\nabla u |=0 \quad \hbox{in} B(0,1)\setminus \Omega,$$ where $\chi_\Omega$ denotes the characteristic function, and the equation is satisfied in the sense of distributions. If the complement of $\Omega$ does not develop cusp singularities at the origin then we prove $\partial \Omega$ is analytic in some small neighborhood of the origin. The result can be modified to yield for more general divergence form operators. As an application of this, then, we obtain the regularity of the boundary of a domain without the Pompeiu property, provided its complement has no cusp singularities.
Caffarelli Luis A.
Karp Lavi
Shahgholian Henrik
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