Calculus of Variations and Geometric Measure Theory

D. Mucci

Strong density results for manifold valued fractional Sobolev maps

created by mucci on 03 Jun 2022


Submitted Paper

Inserted: 3 jun 2022
Last Updated: 3 jun 2022

Pages: 26
Year: 2022


We consider fractional Sobolev classes $W^{s,p}$ of maps defined in high dimensional domains and with values into compact smooth manifolds. The problem of strong density of smooth maps for $s$ lower than one is discussed. An equivalent energy convergence defined through extensions in suitable weighted Sobolev spaces is exploited to obtain a new proof of the density of maps with ''small" singular set. Moreover, a homotopy-type property is analyzed, yielding to a characterization of approximable maps through topological arguments. We then focus on maps taking values into high dimensional spheres, where homological tools allow to describe the singular set. For suitable values of the product $sp$, in fact, strong density of smooth maps is characterized by the triviality of the current of the singularities.

Keywords: weighted Sobolev spaces, fractional Sobolev spaces, Singularities, maps between manifolds