Calculus of Variations and Geometric Measure Theory
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D. Mucci

The relaxed energy of fractional Sobolev maps with values into the circle

created by mucci on 30 Dec 2021

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Submitted Paper

Inserted: 30 dec 2021
Last Updated: 30 dec 2021

Pages: 34
Year: 2021

Abstract:

We deal with the weak sequential density of smooth maps in the fractional Sobolev classes of $W^{s,p}$ maps in high dimension domains and with values into the circle. When $s$ is lower than one, using interpolation theory we introduce a natural energy in terms of optimal extensions on suitable weighted Sobolev spaces. The relaxation problem is then discussed in terms of Cartesian currents. When $sp=1$, the energy gap in the relaxed functional is always finite and is given by the minimal connection of the singularities times an energy weight, obtained through a minimum problem for one dimensional $W^{1/p,p}$ maps with degree one. When $sp>1$, instead, concentration on codimension one sets needs unbounded energy. We finally treat the case where $s$ is greater than one, obtaining an almost complete picture.

Keywords: relaxation, weighted Sobolev spaces, minimal connections, fractional Sobolev spaces, Cartesian currents


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