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

ESAIM: Control, Optimisation and Calculus of Variations (2008)

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

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topMucci, Domenico. "Relaxation of isotropic functionals with linear growth defined on manifold constrained Sobolev mappings." ESAIM: Control, Optimisation and Calculus of Variations 15.2 (2008): 295-321. <http://eudml.org/doc/90915>.

@article{Mucci2008,

abstract = {
In this paper we study the lower semicontinuous envelope with respect to
the L1-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 $\{\cal Y\}$ of $\{\mathbb R\}^\{N\}$.
},

author = {Mucci, Domenico},

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

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

language = {eng},

month = {3},

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/90915},

volume = {15},

year = {2008},

}

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

DA - 2008/3//

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 L1-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 ${\cal Y}$ of ${\mathbb R}^{N}$.

LA - eng

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

UR - http://eudml.org/doc/90915

ER -

## References

top- R. Alicandro and C. Leone, 3D-2D asymptotic analysis for micromagnetic thin films. ESAIM: COCV6 (2001) 489–498. Zbl0989.35009
- 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. Zbl1138.49017
- L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Math. Monographs. Oxford (2000). Zbl0957.49001
- F. Bethuel, The approximation problem for Sobolev maps between manifolds. Acta Math.167 (1992) 153–206.
- B. Dacorogna, I. Fonseca, J. Malý and K. Trivisa, Manifold constrained variational problems. Calc. Var.9 (1999) 185–206. Zbl0935.49006
- F. Demengel and R. Hadiji, Relaxed energies for functionals on ${W}^{1,1}({B}^{n},{\mathbb{S}}^{1})$. Nonlinear Anal.19 (1992) 625–641. Zbl0799.46038
- H. Federer, Geometric measure theory, Grundlehren math. Wissen.153. Springer, Berlin (1969). Zbl0176.00801
- 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. Zbl0788.49039
- 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. Zbl0794.49012
- E. Gagliardo, Caratterizzazione delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili. Rend. Sem. Mat. Univ. Padova27 (1957) 284–305. Zbl0087.10902
- 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. Zbl1150.49020
- 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).
- 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. Zbl1150.49021
- M. Giaquinta and D. Mucci, Relaxation results for a class of functionals with linear growth defined on manifold constrained mappings. Journal of Convex Analysis15 (2008) (online). Zbl1204.49010
- 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. Zbl0810.49040
- 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). Zbl0914.49001
- P.M. Mariano and G. Modica, Ground states in complex bodies. ESAIM: COCV (to appear). Zbl1161.74006
- Y.G. Reshetnyak, Weak convergence of completely additive vector functions on a set. Siberian Math. J.9 (1968) 1039–1045. Zbl0176.44402

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