Title

On the geometry of p-harmonic mappings

Date of Award

2008

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Keywords

p-harmonic, Calculus of variations, Nonlinear PDEs

Subject Categories

Mathematics | Physical Sciences and Mathematics

Abstract

The main subject of this dissertation is the geometry of p -harmonic mappings and related topics. The class of quasiradial mappings in the plane is introduced and their basic properties are investigated. We discuss generating p -harmonic functions and mappings via compositions and observe that in several cases only composition with an affine function produces new p -harmonic functions or mappings. This illustrates the rigidity of p -harmonic mappings.

The variational interpretation of p -harmonics leads to several problems and generalizations. We present radial p -harmonics on ring domains and show solvability of the Dirichlet problem. Existence of four fundamental solutions is proven and their geometric properties are presented. We generalize the p -Dirichlet energy to the weighted case, free-Lagrangians are discussed as well. Next we investigate homeomorphisms with finite distortion and relate them to p -harmonic mappings. The planar Grötzsch problem is extended to higher dimensions and exposed in connections with polyconvex, quasiconvex and rank-one convex functionals. We define the Grötzsch property for energy functionals and show that it holds for a wide class of polyconvex energy functionals.

To every p -harmonic mapping in the plane with p ≥ 2 there corresponds a quasilinear system of first order PDE's which couples the complex gradients of the coordinate functions of the mapping. The ellipticity of such system is proved. A relation between planar quasiregular mappings and p -harmonic mappings is discussed. The p -harmonic conjugate problem is stated.

We finish the exposition with some open problems in the geometry of nonlinear systems.

Access

Surface provides description only. Full text is available to ProQuest subscribers. Ask your Librarian for assistance.

http://libezproxy.syr.edu/login?url=http://proquest.umi.com/pqdweb?did=1597603891&sid=1&Fmt=2&clientId=3739&RQT=309&VName=PQD

Link to Full Text

Share

COinS