# 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

## Access Full Article

top## Abstract

top## How to cite

topAcerbi, 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

top- [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. Zbl0645.58015MR1032767
- [2] F. Bethuel, The approximation problem for Sobolev maps between manifolds. Acta Math.167 (1992) 153–206. Zbl0756.46017MR1120602
- [3] H. Brezis, J.M. Coron and E.H. Lieb, Harmonic maps with defects. Comm. Math. Phys.107 (1986) 649–705. Zbl0608.58016MR868739
- [4] H. Federer, Geometric measure theory, Grundlehren Math. Wissen. 153. Springer, New York (1969). Zbl0176.00801MR257325
- [5] H. Federer and W. Fleming, Normal and integral currents. Annals of Math.72 (1960) 458–520. Zbl0187.31301MR123260
- [6] M. Giaquinta and G. Modica, On sequences of maps with equibounded energies. Calc. Var.12 (2001) 213–222. Zbl1013.49030MR1825872
- [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). Zbl0914.49001MR1645086
- [8] M. Giaquinta and D. Mucci, Density results relative to the Dirichlet energy of mappings into a manifold. Comm. Pure Appl. Math.59 (2006) 1791–1810. Zbl1115.49012MR2257861
- [9] M. Giaquinta and D. Mucci, Maps into manifolds and currents : area and W1,2-, W1/2-, BV-energies, Edizioni della Normale. C.R.M. Series, Sc. Norm. Sup. Pisa (2006). Zbl1111.49001MR2262657
- [10] J. Sacks and K. Uhlenbeck, The existence of minimal immersions of 2-spheres. Annals of Math.113 (1981) 1–24. Zbl0462.58014MR604040
- [11] R. Schoen and K. Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps. J. Diff. Geom.18 (1983) 253–268. Zbl0547.58020MR710054
- [12] L. Simon, Lectures on geometric measure theory, Proc. of the Centre for Math. Analysis 3. Australian National University, Canberra (1983). Zbl0546.49019MR756417
- [13] U. Tarp-Ficenc, On the minimizers of the relaxed energy functionals of mappings from higher dimensional balls into S2. Calc. Var. Partial Differential Equations23 (2005) 451–467. Zbl1074.58006MR2153033
- [14] E.G. Virga, Variational theories for liquid crystals, Applied Mathematics and Mathematical Computation 8. Chapman & Hall, London (1994). Zbl0814.49002MR1369095

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.