Sharp solvability criteria for Dirichlet problems of mean curvature type in Riemannian manifolds: non-existence results

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 {{,mathrm{mathbb {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 times {{,mathrm{mathbb {R}},}}. Precisely, given a mathscr {C}^2 bounded domain Omega in M and a function H = H (x, z) continuous in overline{Omega }times {{,mathrm{mathbb {R}},}} and non-decreasing in the variable z, we prove that the strong Serrin condition(n-1)mathcal {H}_{partial Omega }(y)ge nsup limits _{zin {{,mathrm{mathbb {R}},}}}left| H(y,z) right| forall yin partial Omega , 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.