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

Abstract

The unifying theme of the five articles, [A,B,C,D,E], forming this dissertation is the existence and non-existence of continuous entire non-constant solutions for nonlinear differential operators on a Riemannian manifold M. The existence results of such solutions are proved by studying the asymptotic Dirichlet problem under different assumptions on the geometry of the manifold. Minimal graphic functions are studied in articles [A] and [D]. Article [A] deals with an existence result whereas in [D] we give both existence and non-existence results with respect to the curvature of M. Moreover p-harmonic functions are studied in [D]. Article [B] deals with the existence of A -harmonic functions under similar curvature assumptions as in [A]. In article [C] we study the existence of f-minimal graphs, which are generalisations of usual minimal graphs, and in the article [E] the Killing graphs on warped product manifolds. Before turning to the ideas and results of the research articles, we present some key concepts of the thesis and give a brief history of the development of the asymptotic Dirichlet problem. Due to the similarity of the techniques in [A] and [B], we treat them together in Section 3. Article [C] is treated in Section 4, article [D] in Section 5 and article [E] in Section 6. At the beginning of the Sections 3 – 6 we briefly give the background of the methods and techniques used in the articles.