Dirichlet problems for mean curvature and p-harmonic equations on Cartan-Hadamard manifolds

This dissertation consists of four research articles whose unifying theme is the existence and non-existence of continuous entire non-constant solutions for nonlinear differential operators on Riemannian manifolds. The existence of such solutions depends heavily on the geometry of the manifold and, in the case of complete and simply connected Riemannian manifolds, we prove the existence under assumptions on the sectional curvature.

In the first and fourth article we study the existence of minimal graphic functions by solving the asymptotic Dirichlet problem. Here the idea is to compactify the Cartan-Hadamard manifold by adding an asymptotic boundary and equipping the resulting space with the cone topology. Then one can solve the asymptotic Dirichlet problem i.e. prove the existence of entire solutions with prescribed continuous boundary values on the asymptotic boundary. In the fourth article we prove also a non-existence result by showing that asymptotically non-negative sectional curvature implies uniform gradient estimate for minimal graphic functions with at most linear growth.

The second article deals with the existence of A-harmonic functions and the third article with the existence of f-minimal graphs. In the case of A-harmonic functions, we improve an earlier result of A. Vähäkangas by relaxing the assumption on the curvature upper bound. Here, again, we solve the asymptotic Dirichlet problem in order to get the existence result. We solve the asymptotic Dirichlet problem also for the f-minimal equation, but differing from the other papers, here we consider also the existence in the case of bounded domains.