2005 # 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 elementa…

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“…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…”

confidence: 99%

“…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…”

confidence: 99%

“…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.…”

confidence: 99%

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

confidence: 99%

“…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.…”

confidence: 99%