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

  • Emilio Acerbi (a1), Gianluca Crippa (a1) and Domenico Mucci (a1)


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.



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[1] 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.
[2] Bethuel, F., The approximation problem for Sobolev maps between manifolds. Acta Math. 167 (1992) 153206.
[3] Brezis, H., Coron, J.M. and Lieb, E.H., Harmonic maps with defects. Comm. Math. Phys. 107 (1986) 649705.
[4] H. Federer, Geometric measure theory, Grundlehren Math. Wissen. 153. Springer, New York (1969).
[5] Federer, H. and Fleming, W., Normal and integral currents. Annals of Math. 72 (1960) 458520.
[6] Giaquinta, M. and Modica, G., On sequences of maps with equibounded energies. Calc. Var. 12 (2001) 213222.
[7] 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).
[8] Giaquinta, M. and Mucci, D., Density results relative to the Dirichlet energy of mappings into a manifold. Comm. Pure Appl. Math. 59 (2006) 17911810.
[9] 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).
[10] Sacks, J. and Uhlenbeck, K., The existence of minimal immersions of 2-spheres. Annals of Math. 113 (1981) 124.
[11] Schoen, R. and Uhlenbeck, K., Boundary regularity and the Dirichlet problem for harmonic maps. J. Diff. Geom. 18 (1983) 253268.
[12] L. Simon, Lectures on geometric measure theory, Proc. of the Centre for Math. Analysis 3. Australian National University, Canberra (1983).
[13] 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) 451467.
[14] E.G. Virga, Variational theories for liquid crystals, Applied Mathematics and Mathematical Computation 8. Chapman & Hall, London (1994).



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