Harmonic Maps into Trees and Graphs - Analytical and Numerical Aspects

The main topic of this work is the definition and investigation of a nonlinear energy for maps with values in trees and graphs and the analysis of the corresponding nonlinear Dirichlet problem. The nonlinear energy is defined using a semigroup approach based on Markov kernels and the nonlinear Dirichlet problem is given as a minimizing problem of the nonlinear energy. Conditions for the existence and uniqueness of a solution to the nonlinear Dirichlet problem are presented. A numerical algorithm is developed to solve the nonlinear Dirichlet problem for maps from a two dimensional Euclidean domain into trees. The problem is discretized using a suitable finite element approach and convergence of a corresponding iterative numerical method is proven. Furthermore, for graph targets homotopy problems are analyzed. For particular domain spaces the existence of a minimizer of the nonlinear energy in a given homotopy class is shown. A smooth map f : M → N between Riemannian manifolds is called harmonic if its tension field τ(f) := trace∇(df) vanishes [Jos95]. Well known examples are harmonic functions (N = IR), geodesics (M ⊂ IR) and minimal surfaces. Harmonic maps play an important role in many areas of mathematics, see [EL78], [EL88] for a survey. In the last decade, the study of maps into more general target spaces was developed, e.g. [GS92], [JY93]. Korevaar/Schoen ([KS93], [KS97]) and Jost ([Jos94], [Jos97b]) independently began to develop a theory of harmonic maps into metric spaces of nonpositive curvature in the sense of Alexandrov (briefly: NPC spaces). These developments are based on the fact that a canonical extension of the energy functional can be defined for maps with values in NPC spaces. In the approach by Korevaar/Schoen, the domain space is still a Riemannian manifold. In Jost’s approach, the domain space is a locally compact metric space equipped with an abstract Dirichlet form, replacing the Riemannian manifold equipped with the classical Dirichlet form. Eells/Fuglede study harmonic maps between Riemannian polyhedra in [EF01]. For recent proceedings in the more specific case of maps into Riemannian polyhedra we refer to [Fug01], [Fug03a], [Fug03b]. Picard has investigated harmonic maps into trees [Pic04].