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 Zbl

DOI : 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},
     mrnumber = {2513088},
     zbl = {1167.49015},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2008026/}
}

TY  - JOUR
AU  - Mucci, Domenico
TI  - Relaxation of isotropic functionals with linear growth defined on manifold constrained Sobolev mappings
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2009
SP  - 295
EP  - 321
VL  - 15
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2008026/
DO  - 10.1051/cocv:2008026
LA  - en
ID  - COCV_2009__15_2_295_0
ER  -

%0 Journal Article
%A Mucci, Domenico
%T Relaxation of isotropic functionals with linear growth defined on manifold constrained Sobolev mappings
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2009
%P 295-321
%V 15
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2008026/
%R 10.1051/cocv:2008026
%G en
%F 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/articles/10.1051/cocv:2008026/

Bibliographie
Cité par

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

[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 | Zbl

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

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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

[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 | Zbl

[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 | Zbl

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

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

Cité par Sources :