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Circle-valued maps: bubbles and singularities

Banff International Research Station for Mathematical Innovation and Discovery

We consider maps u : Ω → S1 having some Sobolev regu- larity u ∈ Ws,p. For values of s and p relevant for applications, a) either such maps need not have a phase φ with the same regularity as u b) or the phase φ exists but is not controlled by the norm of u. In case a), “factorization allows to write each such u as u = v w, where v lifts and w is “smoother than u. In the first part of the talk, I will present a new, very simple proof of this result, based on the theory of weighted Sobolev spaces. Case b) occurs only in dimension one. In that case, there is an explicit example of loss of control (a “kink), and it turns out that only kinks lead to loss of control. This translates into a description of weakly convergent sequences which complements the one given by the theory of Cartesian currents. A quantitative result concerning kinks leads to the existence of minimal maps winding once around the unit circle. The common theme of the proofs of the above results is the geometric detection of the energy concentration of manifold-valued maps.

Mathematics; Partial differential equations; Calculus of variations and optimal control; optimization

video/mp4

Author affiliation: Universite Claude-Bernard (Lyon I)

2016-03-10

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/57132

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/