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 $${{\,\mathrm{\mathbb {R}}\,}}^n$$Rn with certain regularity. In this paper we investigate the sharpness of the Serrin condition for the vertical mean curvature equation in the product $$ M^n \times {{\,\mathrm{\mathbb {R}}\,}}$$Mn×R. Precisely, given a $$\mathscr {C}^2$$C2 bounded domain $$\Omega $$Ω in M and a function $$ H = H (x, z) $$H=H(x,z) continuous in $$\overline{\Omega }\times {{\,\mathrm{\mathbb {R}}\,}}$$Ω¯×R and non-decreasing in the variable z, we prove that the strong Serrin condition$$(n-1)\mathcal {H}_{\partial \Omega }(y)\ge n\sup \limits _{z\in {{\,\mathrm{\mathbb {R}}\,}}}\left| H(y,z) \right| \ \forall \ y\in \partial \Omega $$(n-1)H∂Ω(y)≥nsupz∈RH(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.
On complete hypersurfaces of nonnegative sectional curvatures and constant $m$th mean curvature