| Issue |
ESAIM: COCV
Volume 32, 2026
|
|
|---|---|---|
| Article Number | 48 | |
| Number of page(s) | 30 | |
| DOI | https://doi.org/10.1051/cocv/2026031 | |
| Published online | 15 June 2026 | |
A relaxation result for a second order energy of mappings into the sphere
Dipartimento SMFI, Università di Parma,
Parco Area delle Scienze 53/A,
I-43124
Parma,
Italy
* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.
Received:
3
October
2025
Accepted:
6
April
2026
Abstract
A relaxation problem for maps from 3-dimensional domains into the unit 2-sphere is analyzed, 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 a strong approximation result on Cartesian currents.
Mathematics Subject Classification: 49Q15 / 49J45 / 28A75 / 49Q45
Key words: Laplacean / mapping into the sphere / relaxation / currents
© The authors. Published by EDP Sciences, SMAI 2026
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