Regularity of minimizing maps and flows various functionals and targets
Hardt, Robert M.
Doctor of Philosophy
This thesis discusses regularity problems of minimizing maps and flows for various functionals and targets. It consists of four parts: Part 1. Energy Minimizing Mappings into Polyhedra. We prove both the partial interior and complete boundary regularities for maps which minimize energy among all maps into a polyhedron. Part 2. Bubbling Phenomena of Certain Palais-Smale Sequences from Surfaces into General Targets. We show that there is no unaccounted loss of energy for certain Palais-Smale sequences from a surface into a general manifold during the process of bubbling. We also discuss the harmonicity of weak limits of general Palais-Smale sequences. Part 3. Maps Minimizing Convex Functionals between Riemannian Manifolds. We show that any map, which minimizes a uniformly strictly convex $C\sp2$ functionals $F$ among all maps from one manifold to another manifold, has Holder continuous first gradient away from a closed subset with Lebesgue measure zero. Part 4. Existence and Partial Regularity of Weak Flows of Convex Functionals. Assume we are given a $C\sp2$ convex functional $F$, we prove the existence of a weak flow associated to it. We also prove that such a weak flow has Holder continuous spatial gradient away from a closed subset with Lebesgue measure zero.