A variational problem for couples of functions and multifunctions with interaction between leaves

Emilio Acerbi; Gianluca Crippa; Domenico Mucci

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 4, page 1178-1206
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Acerbi, Emilio, Crippa, Gianluca, and Mucci, Domenico. "A variational problem for couples of functions and multifunctions with interaction between leaves." ESAIM: Control, Optimisation and Calculus of Variations 18.4 (2012): 1178-1206. <http://eudml.org/doc/272924>.

@article{Acerbi2012,
abstract = {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.},
author = {Acerbi, Emilio, Crippa, Gianluca, Mucci, Domenico},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {relaxed energies; multifunctions; cartesian currents; Cartesian currents},
language = {eng},
number = {4},
pages = {1178-1206},
publisher = {EDP-Sciences},
title = {A variational problem for couples of functions and multifunctions with interaction between leaves},
url = {http://eudml.org/doc/272924},
volume = {18},
year = {2012},
}

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
PB - EDP-Sciences
VL - 18
IS - 4
SP - 1178
EP - 1206
AB - 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.
LA - eng
KW - relaxed energies; multifunctions; cartesian currents; Cartesian currents
UR - http://eudml.org/doc/272924
ER -

References

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