A new symmetry criterion based on the distance function and applications to PDEʼs

We prove that, if Ω⊂Rn is an open bounded starshaped domain of class C2, the constancy over ∂Ω of the functionφ(y)=∫0λ(y)∏j=1n−1[1−tκj(y)]dt implies that Ω is a ball. Here κj(y) and λ(y) denote respectively the principal curvatures and the cut value of a boundary point y∈∂Ω. We apply this geometric result to different symmetry questions for PDEʼs: an overdetermined system of Monge-Kantorovich type equations (which can be viewed as the limit as p→+∞ of Serrinʼs 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.