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.

For maps from a domain ⊂ R m into a Riemannian manifold N , a functional coming from the norm of a fractional Sobolev space has recently been studied by Da Lio and Rivière. An intrinsically defined functional with a similar behavior also exists, and its first variation can be identified with a Dirichlet-to-Neumann map belonging to the harmonic map problem. The critical points have regularity properties analogous to harmonic maps.Keywords Harmonic map · Dirichlet-to-Neumann map · Regularity Mathematics Subject Classification (2000) 58E20 · 35J50 · 35S99 Harmonic maps between Riemannian manifolds are particularly interesting on a twodimensional domain, because in this case, the problem is invariant under conformal transformations. Using this fact as a motivation, Da Lio and Rivière [2] proposed to study a functional for maps on a one-dimensional domain with related properties. This functional is given in terms of the seminorm belonging to the homogeneous fractional Sobolev spaceḢ 1/2 (R). Such a quantity is very natural from the analytic point of view, and the resulting problem permits the use of tools from harmonic analysis such as a Littlewood-Paley decomposition. The relationship with conformal transformations becomes apparent when the domain is regarded as the boundary of a half-plane. The energy is invariant under the Möbius transformations that map the half-plane onto itself.Communicated by Alexander Isaev. R. Moser ( )

We consider a complex Ginzburg-Landau equation that contains a Schrödinger term and a damping term that is proportional to the time derivative. Given well-prepared initial conditions that correspond to quantized vortices, we establish the vortex motion law until collision time.

Let u be a mapping from a bounded domain S ⊂ ‫ޒ‬ 4 into a compact Riemannian manifold N . Its intrinsic biharmonic energy E 2 (u) is given by the squared L 2 -norm of the intrinsic Hessian of u. We consider weakly converging sequences of critical points of E 2 . Our main result is that the energy dissipation along such a sequence is fully due to energy concentration on a finite set and that the dissipated energy equals a sum over the energies of finitely many entire critical points of E 2 .