On the asymptotic Dirichlet problem for a class of mean curvature type partial differential equations

Abstract

We study the Dirichlet problem for the following prescribed mean curvature PDEwhere Ω is a domain contained in a complete Riemannian manifold M, f : Ω × R → R is a fixed function and ϕ is a given continuous function on ∂Ω. This is done in three parts. In the first one we consider this problem in the most general form, proving the existence of solutions when Ω is a bounded C 2,α domain, under suitable conditions on f , with no restrictions on M besides completeness. In the second part we study the asymptotic Dirichlet problem when M is the hyperbolic space H n and Ω is the whole space. This part uses in an essential way the geometric structure of H n to construct special barriers which resemble the Scherk type solutions of the minimal surface PDE. In the third part one uses these Scherk type graphs to prove the non existence of isolated asymptotic boundary singularities for global solutions of this Dirichlet problem.

Recent rigidity results for graphs with prescribed mean curvature

“…However, we shall see below that a suitably defined Keller-Osserman condition will appear again as soon as has a volume growth faster than polynomial. The guiding example is again = ℍ , where the existence problem for (4) was recently studied in [14].…”

“…In this case = is Killing but not parallel, unless ℎ is constant. For instance, ℍ +1 admits two such warped product decompositions, according to whether has one or two fixed points at infinity: the first can be obtained by isolating the coordinate = in the upper half-space model (14), leading to…”

Survey on the asymptotic Dirichlet problem for the minimal surface equation

“…To finish this section, we mention also the article [5] that appeared after writing the first version of this survey. In [5] Bonorino, Casteras, Klaser, Ripoll and Telichevesky study the asymptotic Dirichlet problem in H n × R with prescribed mean curvature H : H n × R → R depending also on the height variable.…”