We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
Sharp solvability criteria for Dirichlet problems of mean curvature type in Riemannian manifolds: non-existence results.
- Authors
Alvarez, Yunelsy N.; Sa Earp, Ricardo
- Abstract
It is well known that the Serrin condition is a necessary condition for the solvability of the Dirichlet problem for the prescribed mean curvature equation in bounded domains of R n with certain regularity. In this paper we investigate the sharpness of the Serrin condition for the vertical mean curvature equation in the product M n × R . Precisely, given a C 2 bounded domain Ω in M and a function H = H (x , z) continuous in Ω ¯ × R and non-decreasing in the variable z, we prove that the strong Serrin condition (n - 1) H ∂ Ω (y) ≥ n sup z ∈ R H (y , z) ∀ y ∈ ∂ Ω , is a necessary condition for the solvability of the Dirichlet problem in a large class of Riemannian manifolds within which are the Hadamard manifolds and manifolds whose sectional curvatures are bounded above by a positive constant. As a consequence of our results we deduce Jenkins–Serrin and Serrin type sharp solvability criteria.
- Subjects
RIEMANNIAN manifolds; DIRICHLET problem; CURVATURE; HYPERBOLIC spaces; MANIFOLDS (Mathematics)
- Publication
Calculus of Variations & Partial Differential Equations, 2019, Vol 58, Issue 6, pN.PAG
- ISSN
0944-2669
- Publication type
Article
- DOI
10.1007/s00526-019-1649-2