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A variational problem for couples of functions andmultifunctions 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

Abstract

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We discuss a variational problem defined on couples of functions that are constrained totake values into the 2-dimensional unit sphere. The energy functional contains, besidesstandard Dirichlet energies, a non-local interaction term that depends on the distancebetween the gradients of the two functions. Different gradients are preferred or penalizedin this model, in dependence of the sign of the interaction term. In this paper we studythe lower semicontinuity and the coercivity of the energy and we find an explicitrepresentation formula for the relaxed energy. Moreover, we discuss the behavior of theenergy in the case when we consider multifunctions with two leaves rather than couples offunctions.

Keywords

Relaxed energiesmultifunctionsCartesian currents

Information

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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