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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. [email protected]; [email protected]; [email protected]
Gianluca Crippa
Affiliation:
Dipartimento di Matematica dell’Università di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy. [email protected]; [email protected]; [email protected]
Domenico Mucci
Affiliation:
Dipartimento di Matematica dell’Università di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy. [email protected]; [email protected]; [email protected]
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Abstract

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
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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References

F.J. Almgren, W. Browder and E.H. Lieb, Co-area, liquid crystals, and minimal surfaces, in Partial Differential Equations, Lecture Notes in Math. 1306. Springer (1988) 1–22.
Bethuel, F., The approximation problem for Sobolev maps between manifolds. Acta Math. 167 (1992) 153–206. Google Scholar
Brezis, H., Coron, J.M. and Lieb, E.H., Harmonic maps with defects. Comm. Math. Phys. 107 (1986) 649–705. Google Scholar
H. Federer, Geometric measure theory, Grundlehren Math. Wissen. 153. Springer, New York (1969).
Federer, H. and Fleming, W., Normal and integral currents. Annals of Math. 72 (1960) 458–520. Google Scholar
Giaquinta, M. and Modica, G., On sequences of maps with equibounded energies. Calc. Var. 12 (2001) 213–222. Google Scholar
M. Giaquinta, G. Modica and J. Souček, Cartesian currents in the calculus of variations, I, II, Ergebnisse Math. Grenzgebiete (III Ser.) 37, 38. Springer, Berlin (1998).
Giaquinta, M. and Mucci, D., Density results relative to the Dirichlet energy of mappings into a manifold. Comm. Pure Appl. Math. 59 (2006) 1791–1810. Google Scholar
M. Giaquinta and D. Mucci, Maps into manifolds and currents : area and W 1,2-, W 1/2-, BV -energies, Edizioni della Normale. C.R.M. Series, Sc. Norm. Sup. Pisa (2006).
Sacks, J. and Uhlenbeck, K., The existence of minimal immersions of 2-spheres. Annals of Math. 113 (1981) 1–24. Google Scholar
Schoen, R. and Uhlenbeck, K., Boundary regularity and the Dirichlet problem for harmonic maps. J. Diff. Geom. 18 (1983) 253–268. Google Scholar
L. Simon, Lectures on geometric measure theory, Proc. of the Centre for Math. Analysis 3. Australian National University, Canberra (1983).
Tarp-Ficenc, U., On the minimizers of the relaxed energy functionals of mappings from higher dimensional balls into S2. Calc. Var. Partial Differential Equations 23 (2005) 451–467. Google Scholar
E.G. Virga, Variational theories for liquid crystals, Applied Mathematics and Mathematical Computation 8. Chapman & Hall, London (1994).