The $BV$-energy of maps into a manifold : relaxation and density results

Giaquinta, Mariano; Mucci, Domenico

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 4, pp. 483-548.

Let $𝒴$  be a smooth compact oriented riemannian manifoldwithout boundary, and assume that its $1$-homology group has notorsion. Weak limits of graphs of smooth maps $u_k:B^n\to 𝒴$  with equibounded total variation give riseto equivalence classes of cartesian currents in ${\mathrm{cart}}^{1,1}\left(B^n𝒴\right)$  for which we introduce a natural$BV$-energy.Assume moreover that the first homotopy group of  $𝒴$  iscommutative. In any dimension  $n$  we prove that every element $T$  in  ${\mathrm{cart}}^{1,1}\left(B^n𝒴\right)$  can be approximatedweakly in the sense of currents by a sequence of graphs of smoothmaps  $u_k:B^n\to 𝒴$  with total variation converging to the$BV$-energy of  $T$. As a consequence, we characterize the lowersemicontinuous envelope of functions of bounded variations from$B^n$ into $𝒴$.

Classification : 49Q15, 49Q20

[1] L. Ambrosio, Metric space valued functions of bounded variation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (1990), 439-478. | EuDML | Numdam | MR | Zbl

[2] L. Ambrosio, N. Fusco and D. Pallara, “Functions of bounded Variation and Free Discontinuity Problems”, Oxford Math. Monographs, Oxford, 2000. | MR | Zbl

[3] L. Ambrosio and B. Kirchheim, Currents in metric spaces, Acta Math. 185 (2000), 1-80. | MR | Zbl

[4] P. Aviles and Y. Giga, Variational integrals of mappings of bounded variation and their lower semicontinuity, Arch. Ration. Mech. Anal. 115 (1991), 201-255. | MR | Zbl

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

[6] F. Bethuel, J. M. Coron, F. Demengel and F. Helein, A cohomological criterium for density of smooth maps in Sobolev spaces between two manifolds, In: “Nematics, Mathematical and Physical Aspects”, J. M. Coron, J. M. Ghidaglia, F. Helein (eds.), NATO ASI Series C, 332, Kluwer Academic Publishers, Dordrecht, 1991, 15-23. | MR | Zbl

[7] G. Bouchitté and G. Buttazzo, New lower semicontinuity results for nonconvex functionals defined on measures, Nonlinear Anal. 15 (1990), 679-692. | MR | Zbl

[8] H. Brezis, P. Mironescu and A. Ponce, $W^{1,1}$-maps with value into $S^1$, In: “Geometric Analysis of PDE and Several Complex Variables”, S. Chanillo, P. Cordaro, N. Hanges and A. Meziani (eds.), Contemporary Mathematics, 368, American Mathematical Society, Providence, RI, 2005, 69-100. | MR | Zbl

[9] H. Federer, “Geometric Measure Theory”, Grundlehren math. Wissen. 153, Springer, Berlin, 1969. | MR | Zbl

[10] H. Federer, Real flat chains, cochains and variational problems, Indiana Univ. Math. J. 24 (1974), 351-407. | MR | Zbl

[11] 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

[12] M. Giaquinta and G. Modica, On sequences of maps with equibounded energies, Calc. Var. Partial Differential Equations 12 (2001), 213-222. | MR | Zbl

[13] 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

[14] 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

[15] M. Giaquinta and D. Mucci, Weak and strong density results for the Dirichlet energy, J. Eur. Math. Soc. 6 (2004), 95-117. | MR | Zbl

[16] M. Giaquinta and D. Mucci, The Dirichlet energy of mappings from $B^3$ into a manifold: density results and gap phenomenon, Calc. Var. Partial Differential Equations 20 (2004), 367-397. | MR | Zbl

[17] M. Giaquinta and D. Mucci, On sequences of maps into a manifold with equibounded $W^{1/2}$-energies, J. Funct. Anal. 225 (2005), 94-146. | MR | Zbl

[18] B. Hardt and J. Pitts, Solving the Plateau's problem for hypersurfaces without the compactness theorem for integral currents, In: “Geometric Measure Theory and the Calculus of Variations”, W. K. Allard and F. J. Almgren (eds.), Proc. Symp. Pure Math. 44. Amer. Math. Soc., Providence, 1996, 255-295. | MR | Zbl

[19] R. Ignat, The space $BV(S^2;S^1)$: minimal connections and optimal liftings Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 283-302. | Numdam | MR | Zbl

[20] M. R. Pakzad and T. Rivière, Weak density of smooth maps for the Dirichlet energy between manifolds, Geom. Funct. Anal. 13 (2001), 223-257. | MR | Zbl

[21] R. Schoen and K Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps, J. Differential Geom. 18 (1983), 253-268. | MR | Zbl

[22] L. Simon, “Lectures on geometric measure theory”, Proc. C.M.A., Vol. 3, Australian Natl. Univ., Canberra, 1983. | MR | Zbl

[23] B. White, Rectifiability of flat chains, Ann. of Math. (2) 150 (1999), 165-184. | MR | Zbl