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 ohm subset of M be a bounded domain, with O is an element of ohm, and consider the problem Delta(p)u = - 1 in O with u = 0 on partial derivative ohm, where Delta(p) is the p-Laplacian of g. We prove that if the normal derivative partial derivative(v)u of u along the boundary of ohm is a function of d satisfying suitable conditions, then ohm must be a geodesic ball. In particular, our result applies to open balls of R-n equipped with a rotationally symmetric metric of the form g = dt(2) + rho(2)(t) gs, where gS is the standard metric of the sphere.
A remark on an overdetermined problem in riemannian geometry / G. Ciraolo, L. Vezzoni (SPRINGER PROCEEDINGS IN MATHEMATICS & STATISTICS). - In: Geometric Properties for Parabolic and Elliptic PDE's / [a cura di] F. Gazzola, K. Ishige, C. Nitsch,P. Salani. - [s.l] : Springer, 2016. - ISBN 9783319415369. - pp. 87-96 (( Intervento presentato al 4. convegno Italian-Japanese Workshop on Geometric Properties for Parabolic and Elliptic PDE's tenutosi a Palinuro nel 2015 [10.1007/978-3-319-41538-3_6].
A remark on an overdetermined problem in riemannian geometry
G. Ciraolo;
2016
Abstract
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 ohm subset of M be a bounded domain, with O is an element of ohm, and consider the problem Delta(p)u = - 1 in O with u = 0 on partial derivative ohm, where Delta(p) is the p-Laplacian of g. We prove that if the normal derivative partial derivative(v)u of u along the boundary of ohm is a function of d satisfying suitable conditions, then ohm must be a geodesic ball. In particular, our result applies to open balls of R-n equipped with a rotationally symmetric metric of the form g = dt(2) + rho(2)(t) gs, where gS is the standard metric of the sphere.File | Dimensione | Formato | |
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