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

ESAIM: Control, Optimisation and Calculus of Variations (2009)

- Volume: 15, Issue: 2, page 295-321
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topMucci, Domenico. "Relaxation of isotropic functionals with linear growth defined on manifold constrained Sobolev mappings." ESAIM: Control, Optimisation and Calculus of Variations 15.2 (2009): 295-321. <http://eudml.org/doc/245465>.

@article{Mucci2009,

abstract = {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 $\{\mathbb \{R\}\}^\{N\}$ that are constrained to take values into a smooth submanifold $\{\mathcal \{Y\}\}$ of $\{\mathbb \{R\}\}^\{N\}$.},

author = {Mucci, Domenico},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {relaxation; manifold constrain; BV functions; manifold constraints},

language = {eng},

number = {2},

pages = {295-321},

publisher = {EDP-Sciences},

title = {Relaxation of isotropic functionals with linear growth defined on manifold constrained Sobolev mappings},

url = {http://eudml.org/doc/245465},

volume = {15},

year = {2009},

}

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

PB - EDP-Sciences

VL - 15

IS - 2

SP - 295

EP - 321

AB - 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 ${\mathbb {R}}^{N}$ that are constrained to take values into a smooth submanifold ${\mathcal {Y}}$ of ${\mathbb {R}}^{N}$.

LA - eng

KW - relaxation; manifold constrain; BV functions; manifold constraints

UR - http://eudml.org/doc/245465

ER -

## References

top- [1] R. Alicandro and C. Leone, 3D-2D asymptotic analysis for micromagnetic thin films. ESAIM: COCV 6 (2001) 489–498. Zbl0989.35009MR1836053
- [2] R. Alicandro, A. Corbo Esposito and C. Leone, Relaxation in $BV$ of functionals defined on Sobolev functions with values into the unit sphere. J. Convex Anal. 14 (2007) 69–98. Zbl1138.49017MR2310429
- [3] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Math. Monographs. Oxford (2000). Zbl0957.49001MR1857292
- [4] F. Bethuel, The approximation problem for Sobolev maps between manifolds. Acta Math. 167 (1992) 153–206. Zbl0756.46017MR1120602
- [5] B. Dacorogna, I. Fonseca, J. Malý and K. Trivisa, Manifold constrained variational problems. Calc. Var. 9 (1999) 185–206. Zbl0935.49006MR1725201
- [6] F. Demengel and R. Hadiji, Relaxed energies for functionals on ${W}^{1,1}({B}^{n},{\mathbb{S}}^{1})$. Nonlinear Anal. 19 (1992) 625–641. Zbl0799.46038MR1186122
- [7] H. Federer, Geometric measure theory, Grundlehren math. Wissen. 153. Springer, Berlin (1969). Zbl0176.00801MR257325
- [8] I. Fonseca and S. Müller, Relaxation of quasiconvex functionals in $BV(\Omega ,{\mathbb{R}}^{p})$ for integrands $f(x,u,\nabla u)$. Arch. Rat. Mech. Anal. 123 (1993) 1–49. Zbl0788.49039MR1218685
- [9] I. Fonseca and P. Rybka, Relaxation of multiple integrals in the space $BV(\Omega ,{\mathbb{R}}^{p})$. Proc. Royal Soc. Edin. 121A (1992) 321–348. Zbl0794.49012MR1179823
- [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. Zbl0087.10902MR102739
- [11] M. Giaquinta and D. Mucci, The $BV$-energy of maps into a manifold: relaxation and density results. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 5 (2006) 483–548. Zbl1150.49020MR2297721
- [12] 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). Zbl1111.49001MR2262657
- [13] M. Giaquinta and D. Mucci, Erratum and addendum to: The $BV$-energy of maps into a manifold: relaxation and density results. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 6 (2007) 185–194. Zbl1150.49021MR2341520
- [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). Zbl1204.49010MR2489611
- [15] M. Giaquinta, G. Modica and J. Souček, Variational problems for maps of bounded variations with values in ${\mathbb{S}}^{1}$. Calc. Var. 1 (1993) 87–121. Zbl0810.49040MR1261719
- [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). Zbl0914.49001MR1645086
- [17] P.M. Mariano and G. Modica, Ground states in complex bodies. ESAIM: COCV (to appear). Zbl1161.74006MR2513091
- [18] Y.G. Reshetnyak, Weak convergence of completely additive vector functions on a set. Siberian Math. J. 9 (1968) 1039–1045. Zbl0176.44402