*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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