Serrin’s overdetermined problem and constant mean curvature surfaces

Abstract

For all $N\ge 9$, we find smooth entire epigraphs in ${\mathbb{R}}^N$, namely, smooth domains of the form $\Omega :=\{x\in {\mathbb{R}}^N|x_N>F(x_1,\dots ,x_{N-1})\}$, which are not half-spaces and in which a problem of the form $\Delta u+f\left(u\right)=0$ in $\Omega$ has a positive, bounded solution with $0$ Dirichlet boundary data and constant Neumann boundary data on $\partial \Omega$. This answers negatively for large dimensions a question by Berestycki, Caffarelli, and Nirenberg. In 1971, Serrin proved that a bounded domain where such an overdetermined problem is solvable must be a ball, in analogy to a famous result by Alexandrov that states that an embedded compact surface with constant mean curvature (CMC) in Euclidean space must be a sphere. In lower dimensions we succeed in providing examples for domains whose boundary is close to large dilations of a given CMC surface where Serrin’s overdetermined problem is solvable.