Calculus of Variations and Geometric Measure Theory

D. Mucci

A relaxation result for a second order energy of mappings into the sphere

created by mucci on 09 Mar 2023


Submitted Paper

Inserted: 9 mar 2023
Last Updated: 9 mar 2023

Year: 2023


A relaxation problem for maps from n-dimensional domains into the unit 2-sphere is analysed, the energy being given in the smooth case by the integral of the modulus of the Laplacean vector. For second order Sobolev maps, a complete explicit formula of the relaxed energy is obtained. Our proof is based on the following results: minimal energy computation of maps with fixed degree, Dipole-like problems, density of maps with small singular sets, lower semicontinuity of the extended energy, and strong approximation properties on Cartesian currents.

Keywords: relaxation, currents, Laplacean, mapping into the sphere