Relaxation of isotropic functionals with linear growth defined on manifold constrained Sobolev mappings

Mucci, Domenico

doi:10.1051/cocv:2008026

Mucci, Domenico

ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 2, pp. 295-321.

Résumé

In this paper we study the lower semicontinuous envelope with respect to the $L^{1}$ -topology of a class of isotropic functionals with linear growth defined on mappings from the $n$ -dimensional ball into $ℝ^{N}$ that are constrained to take values into a smooth submanifold $𝒴$ of $ℝ^{N}$ .

MR 2513088 | Zbl 1167.49015

DOI : https://doi.org/10.1051/cocv:2008026
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 = {http://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/

Bibliographie

[1] R. Alicandro and C. Leone, 3D-2D asymptotic analysis for micromagnetic thin films. ESAIM: COCV 6 (2001) 489-498. | Numdam | MR 1836053 | Zbl 0989.35009

[2] R. Alicandro, A. Corbo Esposito and C. Leone, Relaxation in $B V$ of functionals defined on Sobolev functions with values into the unit sphere. J. Convex Anal. 14 (2007) 69-98. | MR 2310429 | Zbl 1138.49017

[3] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Math. Monographs. Oxford (2000). | MR 1857292 | Zbl 0957.49001

[4] F. Bethuel, The approximation problem for Sobolev maps between manifolds. Acta Math. 167 (1992) 153-206. | MR 1120602 | Zbl 0756.46017

[5] B. Dacorogna, I. Fonseca, J. Malý and K. Trivisa, Manifold constrained variational problems. Calc. Var. 9 (1999) 185-206. | MR 1725201 | Zbl 0935.49006

[6] F. Demengel and R. Hadiji, Relaxed energies for functionals on $W^{1, 1} (B^{n}, 𝕊^{1})$ . Nonlinear Anal. 19 (1992) 625-641. | MR 1186122 | Zbl 0799.46038

[7] H. Federer, Geometric measure theory, Grundlehren math. Wissen. 153. Springer, Berlin (1969). | MR 257325 | Zbl 0176.00801

[8] I. Fonseca and S. Müller, Relaxation of quasiconvex functionals in $B V (Ω, ℝ^{p})$ for integrands $f (x, u, \nabla u)$ . Arch. Rat. Mech. Anal. 123 (1993) 1-49. | MR 1218685 | Zbl 0788.49039

[9] I. Fonseca and P. Rybka, Relaxation of multiple integrals in the space $B V (Ω, ℝ^{p})$ . Proc. Royal Soc. Edin. 121A (1992) 321-348. | MR 1179823 | Zbl 0794.49012

[10] E. Gagliardo, Caratterizzazione delle tracce sulla frontiera relative ad alcune classi di funzioni in $n$ variabili. Rend. Sem. Mat. Univ. Padova 27 (1957) 284-305. | Numdam | MR 102739 | Zbl 0087.10902

[11] M. Giaquinta and D. Mucci, The $B V$ -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

[12] M. Giaquinta and D. Mucci, Maps into manifolds and currents: area and $W^{1, 2}$ -, $W^{1 / 2}$ -, $B V$ -energies. Edizioni della Normale, C.R.M. Series, Sc. Norm. Sup. Pisa (2006). | MR 2262657 | Zbl 1111.49001

[13] M. Giaquinta and D. Mucci, Erratum and addendum to: The $B V$ -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

[14] M. Giaquinta and D. Mucci, Relaxation results for a class of functionals with linear growth defined on manifold constrained mappings. Journal of Convex Analysis 15 (2008) (online). | MR 2489611

[15] M. Giaquinta, G. Modica and J. Souček, Variational problems for maps of bounded variations with values in $𝕊^{1}$ . Calc. Var. 1 (1993) 87-121. | MR 1261719 | Zbl 0810.49040

[16] 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). | MR 1645086 | Zbl 0914.49001

[17] P.M. Mariano and G. Modica, Ground states in complex bodies. ESAIM: COCV (to appear). | Numdam | MR 2513091 | Zbl 1161.74006

[18] Y.G. Reshetnyak, Weak convergence of completely additive vector functions on a set. Siberian Math. J. 9 (1968) 1039-1045. | Zbl 0176.44402