Minimizers of the W-1,W-1-energy of S-1-valued maps with prescribed singularities. Do they exist?

摘要

The paper is concerned with the least W-1,W-1-energy required to produce maps from a domain Omega subset of R-2 with values into S-1 having prescribed singularities (a(i))(1 = i = k). The value of the infimum has been known for a long time and corresponds to the length of minimal configurations connecting the points (a(i)) between themselves and/or to the boundary. We tackle here the question whether the infimum of this W-1,W-1-energy is achieved. This natural topic turns out to be delicate and we have a complete answer only when k = 1. The bottom line for k = 1 is that the infimum is "rarely" achieved. As a "substitute", we give a full description of the asymptotic behavior of all minimizing sequences and show that they "concentrate" along "convex combinations" of minimal configurations. (C) 2018 Elsevier Ltd. All rights reserved.