A new symmetry criterion based on the distance function and applications to PDE's

Abstract

We prove that, if $\Omega \subset {\mathbb{R}}^n$ is an open bounded starshaped domain of class $C^2$, the constancy over $\partial \Omega$ of the function $$\phi (y)={\int }_0^{\lambda (y)}\prod _{j=1}^{n-1}[1-t{\kappa }_j(y)]dt$$ implies that $\Omega$ is a ball. Here $k_j(y)$ and $\lambda (y)$ denote respectively the principal curvatures and the cut value of a boundary point $y\in \partial \Omega$. We apply this geometric result to different symmetry questions for PDE's: an overdetermined system of Monge-Kantorovich type equations (which can be viewed as the limit as $p\to +\infty$ of Serrin's symmetry problem for the $p$-Laplacian), and equations in divergence form whose solutions depend only on the distance from the boundary in some subset of their domain.

17 pages

References in corpus (1)

• A nonhomogeneous boundary value problem in mass transfer theory

Cited by in corpus (1)

• On the characterization of some classes of proximally smooth sets