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

Published online by Cambridge University Press:  16 January 2012

Emilio Acerbi ,
Gianluca Crippa  and
Domenico Mucci
Emilio Acerbi
Affiliation:
Dipartimento di Matematica dell’Università di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy. emilio.acerbi@unipr.it; gianluca.crippa@unipr.it; domenico.mucci@unipr.it
Gianluca Crippa
Affiliation:
Dipartimento di Matematica dell’Università di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy. emilio.acerbi@unipr.it; gianluca.crippa@unipr.it; domenico.mucci@unipr.it
Domenico Mucci
Affiliation:
Dipartimento di Matematica dell’Università di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy. emilio.acerbi@unipr.it; gianluca.crippa@unipr.it; domenico.mucci@unipr.it
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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.

Keywords

Relaxed energiesmultifunctionsCartesian currents
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 4 , October 2012 , pp. 1178 - 1206
Copyright
© EDP Sciences, SMAI, 2012

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