Calculus of Variations and Geometric Measure Theory

D. Mucci

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

created by mucci on 30 Dec 2021


Submitted Paper

Inserted: 30 dec 2021
Last Updated: 30 dec 2021

Pages: 34
Year: 2021


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