Partial Regularity for Harmonic Maps and Related Problems

Abstract

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.

Solving variational problems and partial differential equations that map between manifolds via the closest point method

“…As we saw in Section 2, the term ∆ M u can be replaced by ∆u(cp M ) using the CPM M . Furthermore, the projection operator Π T u N equals the Jacobian of the closest point function, J cp N (u) , for u ∈ N [29,31]. Applying these replacements gives the embedding gradient descent flow…”

“…Harmonic maps [31,32,33] are important in many applications such as texture mapping [8], regularization of brain images [9], and colour image enhancement [5]. Considerable research on the theory of harmonic maps has also been carried out, starting with the work of Fuller [34] in 1954 and the more general theory by Eells and Sampson [35] in 1964.…”

“…Mémoli et al [27] derived the Euler-Lagrange equations for (2) in terms of the level set representation of N under the assumption that M is flat and open. The same calculation is carried out by Moser [31] in terms of the closest point representation of N. There, the closest point function is called the nearest point projection and is used to prove regularity results of harmonic maps (see Chapter 3 of [31]). The Euler-Lagrange equations corresponding to (2), assuming M is flat and open, are…”

Curvature-stabilized skyrmions with angular momentum

“…Smoothness follows from well-established methods from the regularity theory of harmonic maps from surfaces, see e.g. [22].…”

On Uniqueness for the Harmonic Map Heat Flow in Supercritical Dimensions

“…These last results are optimal: indeed, for data that are large in L 1 but not continuous, uniqueness is lost, and examples of nonunique solutions can be constructed (see Coron [13] and Bethuel-Coron-Ghidaglia-Soyeur [2]). For these examples, however, uniqueness can be salvaged if one requires that solutions satisfy the monotonicity formula (itself essentially a consequence of the local energy inequality; see Moser [39]). This led Struwe [48] to ask whether this criterion could indeed imply uniqueness.…”