We discuss a variational problem defined on couples of functions that are constrained to take values into the 2-dimensional unit sphere. The energy functional contains, besides standard Dirichlet energies, a non-local interaction term that depends on the distance between the gradients of the two functions. Different gradients are preferred or penalized in this model, in dependence of the sign of the interaction term. In this paper we study the lower semicontinuity and the coercivity of the energy and we find an explicit representation formula for the relaxed energy. Moreover, we discuss the behavior of the energy in the case when we consider multifunctions with two leaves rather than couples of functions.
Mots-clés : relaxed energies, multifunctions, cartesian currents
@article{COCV_2012__18_4_1178_0,
author = {Acerbi, Emilio and Crippa, Gianluca and Mucci, Domenico},
title = {A variational problem for couples of functions and multifunctions with interaction between leaves},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1178--1206},
publisher = {EDP-Sciences},
volume = {18},
number = {4},
year = {2012},
doi = {10.1051/cocv/2011195},
mrnumber = {3019477},
zbl = {1259.49067},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2011195/}
}
TY - JOUR AU - Acerbi, Emilio AU - Crippa, Gianluca AU - Mucci, Domenico TI - A variational problem for couples of functions and multifunctions with interaction between leaves JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2012 SP - 1178 EP - 1206 VL - 18 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2011195/ DO - 10.1051/cocv/2011195 LA - en ID - COCV_2012__18_4_1178_0 ER -
%0 Journal Article %A Acerbi, Emilio %A Crippa, Gianluca %A Mucci, Domenico %T A variational problem for couples of functions and multifunctions with interaction between leaves %J ESAIM: Control, Optimisation and Calculus of Variations %D 2012 %P 1178-1206 %V 18 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2011195/ %R 10.1051/cocv/2011195 %G en %F COCV_2012__18_4_1178_0
Acerbi, Emilio; Crippa, Gianluca; Mucci, Domenico. A variational problem for couples of functions and multifunctions with interaction between leaves. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 4, pp. 1178-1206. doi : 10.1051/cocv/2011195. http://www.numdam.org/articles/10.1051/cocv/2011195/
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