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
modified on 14 Feb 2025

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Submitted Paper

Inserted: 9 mar 2023
Last Updated: 14 feb 2025

Year: 2023
Notes:

Revised version.


Abstract:

A relaxation problem for maps from 3-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, lower semicontinuity of the extended energy, and strong approximation properties on Cartesian currents.

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


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