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

Mucci, Domenico

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

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 ${ℝ}^N$.

[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 $BV$ 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 $BV(\Omega ,{ℝ}^p)$ 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 $BV(\Omega ,{ℝ}^p)$. 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 $BV$-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 $W^{1,2}$-, $W^{1/2}$-, $BV$-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 $BV$-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