PrefaceThis monograph contains material from several research papers and lectures I gave in Bonn, Leipzig, New York, and Fribourg on various occasions, all of them about different aspects of the same problem. In an attempt to make the work nearly self-contained, I also included many additional paragraphs and most of the proofs of the auxiliary results. It is assumed, however, that the reader is familiar with the basic theory of linear elliptic and parabolic partial differential equations, and with the elementary notions of Riemannian geometry.The aim of the book is to explain the methods that have been developed in the last decades to prove partial regularity for harmonic maps, and also to show how these methods can be extended to related problems. This includes perturbations of the harmonic map problem as well as associated parabolic problems. Both types may be of interest in applications from physics or possibly other sciences.

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We study a variational model from micromagnetics involving a nonlocal Ginzburg-Landau type energy for S 1 -valued vector fields. These vector fields form domain walls, called Néel walls, that correspond to one-dimensional transitions between two directions within the unit circle S 1 . Due to the nonlocality of the energy, a Néel wall is a two length scale object, comprising a core and two logarithmically decaying tails. Our aim is to determine the energy differences leading to repulsion or attraction between Néel walls. In contrast to the usual Ginzburg-Landau vortices, we obtain a renormalised energy for Néel walls that shows both a tail-tail interaction and a core-tail interaction. This is a novel feature for Ginzburg-Landau type energies that entails attraction between Néel walls of the same sign and repulsion between Néel walls of opposite signs.

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We consider the level set formulation of the inverse mean curvature flow. We establish a connection to the problem of p-harmonic functions and give a new proof for the existence of weak solutions.