In this paper we study the lower semicontinuous envelope with respect to the -topology of a class of isotropic functionals with linear growth defined on mappings from the -dimensional ball into that are constrained to take values into a smooth submanifold of .
Classification : 49J45, 49Q20
Mots clés : relaxation, manifold constrain, BV functions
@article{COCV_2009__15_2_295_0, author = {Mucci, Domenico}, title = {Relaxation of isotropic functionals with linear growth defined on manifold constrained Sobolev mappings}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {295--321}, publisher = {EDP-Sciences}, volume = {15}, number = {2}, year = {2009}, doi = {10.1051/cocv:2008026}, zbl = {1167.49015}, mrnumber = {2513088}, language = {en}, url = {www.numdam.org/item/COCV_2009__15_2_295_0/} }
Mucci, Domenico. Relaxation of isotropic functionals with linear growth defined on manifold constrained Sobolev mappings. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 2, pp. 295-321. doi : 10.1051/cocv:2008026. http://www.numdam.org/item/COCV_2009__15_2_295_0/
[1] 3D-2D asymptotic analysis for micromagnetic thin films. ESAIM: COCV 6 (2001) 489-498. | Numdam | MR 1836053 | Zbl 0989.35009
and ,[2] Relaxation in of functionals defined on Sobolev functions with values into the unit sphere. J. Convex Anal. 14 (2007) 69-98. | MR 2310429 | Zbl 1138.49017
, and ,[3] Functions of bounded variation and free discontinuity problems, Oxford Math. Monographs. Oxford (2000). | MR 1857292 | Zbl 0957.49001
, and ,[4] The approximation problem for Sobolev maps between manifolds. Acta Math. 167 (1992) 153-206. | MR 1120602 | Zbl 0756.46017
,[5] Manifold constrained variational problems. Calc. Var. 9 (1999) 185-206. | MR 1725201 | Zbl 0935.49006
, , and ,[6] Relaxed energies for functionals on . Nonlinear Anal. 19 (1992) 625-641. | MR 1186122 | Zbl 0799.46038
and ,[7] Geometric measure theory, Grundlehren math. Wissen. 153. Springer, Berlin (1969). | MR 257325 | Zbl 0176.00801
,[8] Relaxation of quasiconvex functionals in for integrands . Arch. Rat. Mech. Anal. 123 (1993) 1-49. | MR 1218685 | Zbl 0788.49039
and ,[9] Relaxation of multiple integrals in the space . Proc. Royal Soc. Edin. 121A (1992) 321-348. | MR 1179823 | Zbl 0794.49012
and ,[10] Caratterizzazione delle tracce sulla frontiera relative ad alcune classi di funzioni in variabili. Rend. Sem. Mat. Univ. Padova 27 (1957) 284-305. | Numdam | MR 102739 | Zbl 0087.10902
,[11] The -energy of maps into a manifold: relaxation and density results. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 5 (2006) 483-548. | Numdam | MR 2297721 | Zbl 1150.49020
and ,[12] Maps into manifolds and currents: area and -, -, -energies. Edizioni della Normale, C.R.M. Series, Sc. Norm. Sup. Pisa (2006). | MR 2262657 | Zbl 1111.49001
and ,[13] Erratum and addendum to: The -energy of maps into a manifold: relaxation and density results. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 6 (2007) 185-194. | MR 2341520 | Zbl 1150.49021
and ,[14] Relaxation results for a class of functionals with linear growth defined on manifold constrained mappings. Journal of Convex Analysis 15 (2008) (online). | MR 2489611 | Zbl pre05493302
and ,[15] Variational problems for maps of bounded variations with values in . Calc. Var. 1 (1993) 87-121. | MR 1261719 | Zbl 0810.49040
, and ,[16] Cartesian currents in the calculus of variations, I, II. Ergebnisse Math. Grenzgebiete (III Ser.) 37, 38. Springer, Berlin (1998). | MR 1645086 | Zbl 0914.49001
, and ,[17] Ground states in complex bodies. ESAIM: COCV (to appear). | Numdam | MR 2513091 | Zbl 1161.74006
and ,[18] Weak convergence of completely additive vector functions on a set. Siberian Math. J. 9 (1968) 1039-1045. | Zbl 0176.44402
,