期刊
JOURNAL OF FUNCTIONAL ANALYSIS
卷 217, 期 2, 页码 249-279出版社
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jfa.2003.12.007
关键词
degree of VMO maps; noncompact minimizing sequences; concentration phenomena
类别
We consider, for maps in H-1/2(S-1; S-1), a family of (semi)norms equivalent to the standard one. We ask whether, for such a norm, there is some map in H-1/2(S-1; S-1) of prescribed topological degree equal to 1 and minimal norm. In general, the answer is no, due to concentration phenomena. The existence of a minimal map is sensitive to small perturbations of the norm. We derive a sufficient condition for the existence of minimal maps. In particular, we prove that, for every given norm, there are arbitrarily small perturbations of it for which the minimum is attained. In case there is no minimizer, we determine the asymptotic behavior of minimizing sequences. We prove that, for such minimizing sequences, the energy concentrates near a point of S-1. We describe this concentration in terms of bubbling-off of circles. (C) 2004 Elsevier Inc. All rights reserved.
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