TY - CONF

T1 - A remark on an overdetermined problem in riemannian geometry

AU - Ciraolo, Giulio

AU - Vezzoni, Luigi

PY - 2016

Y1 - 2016

N2 - Let (M, g) be a Riemannian manifold with a distinguished point O and assume that the geodesic distance d from O is an isoparametric function. Let Ω ⊂ M be a bounded domain, with O ∈ Ω, and consider the problem Δpu = −1 in Ω with u = 0on∂Ω, where Δp is the p-Laplacian of g. We prove that if the normal derivative ∂νu of u along the boundary of Ω is a function of d satisfying suitable conditions, then Ω must be a geodesic ball. In particular, our result applies to open balls of Rn equipped with a rotationally symmetric metric of the form g = dt2 + ρ2 (t) gS, where gS is the standard metric of the sphere.

AB - Let (M, g) be a Riemannian manifold with a distinguished point O and assume that the geodesic distance d from O is an isoparametric function. Let Ω ⊂ M be a bounded domain, with O ∈ Ω, and consider the problem Δpu = −1 in Ω with u = 0on∂Ω, where Δp is the p-Laplacian of g. We prove that if the normal derivative ∂νu of u along the boundary of Ω is a function of d satisfying suitable conditions, then Ω must be a geodesic ball. In particular, our result applies to open balls of Rn equipped with a rotationally symmetric metric of the form g = dt2 + ρ2 (t) gS, where gS is the standard metric of the sphere.

KW - Comparison principle

KW - Isoparametric functions

KW - Mathematics (all)

KW - Overdetermined PDE

KW - Riemannian Geometry

KW - Rotationally symmetric spaces

KW - Comparison principle

KW - Isoparametric functions

KW - Mathematics (all)

KW - Overdetermined PDE

KW - Riemannian Geometry

KW - Rotationally symmetric spaces

UR - http://hdl.handle.net/10447/201409

UR - http://www.springer.com/series/10533

M3 - Other

SP - 87

EP - 96

ER -