Functions with prescribed singularities

G. Alberti1, S. Baldo2, G. Orlandi3
1Dipartimento di Matematica, Università di Pisa, Pisa, Italy
2Dipartimento di Matematica, Università di Trento, Povo, Italy
3Dipartimento di Informatica, Università di Verona, Verona, Italy

Tóm tắt

The distributional k-dimensional Jacobian of a map u in the Sobolev space W 1,k-1 which takes values in the the sphere S k-1 can be viewed as the boundary of a rectifiable current of codimension k carried by (part of) the singularity of u which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary M of codimension k can be realized as Jacobian of a Sobolev map valued in S k-1. In case M is polyhedral, the map we construct is smooth outside M plus an additional polyhedral set of lower dimension, and can be used in the constructive part of the proof of a Γ-convergence result for functionals of Ginzburg-Landau type, as described in [2].

Tài liệu tham khảo

Alberti, G.: Un risultato di convergenza variazionale per funzionali di tipo Ginzburg-Landau in dimensione qualunque (A variational convergence result for functionals of Ginzburg-Landau type in any dimension). Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 4, 289–310 (2001) Alberti, G., Baldo, S., Orlandi, G.: Variational convergence for functionals of Ginzburg-Landau type. Preprint 2002 Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. Oxford: Oxford Science Publications 1999 Baldo, S., Orlandi, G.: A note on the Hodge theory for functionals with linear growth. Manuscr. Math. 97, 453–467 (1998) Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1977) Bethuel, F.: The approximation problem for Sobolev maps between two manifolds. Acta Math. 167, 153–206 (1991) Bethuel, F., Coron, J.-M., Demengel, F., Hélein, F.: A cohomological criterion for density of smooth maps in Sobolev spaces between two manifolds. Nematics. Mathematical and physical aspects (Orsay, 1990), 15–23, ed. by J.-M. Coron et al. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 332. Dordrecht: Kluwer Academic Publishers 1991 Bethuel, F., Demengel, F.: Extensions of Sobolev mappings between manifolds. Calc. Var. Partial Differ. Equ. 3, 375–391 (1995) Bochnak, J., Kucharz, W.: Complete intersections in differential topology and analytic geometry. Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (7) 10, 1019–1041 (1996) Bott, R., Tu, L.W.: Differential forms in algebraic topology. Graduate Texts in Mathematics, 82. New York, Berlin: Springer 1982 Bourgain, J., Brezis, H., Mironescu, P.: Lifting in Sobolev spaces. J. Anal. Math. 80, 37–86 (2000) Bourgain, J., Brezis, H., Mironescu, P.: On the structure of the Sobolev space H 1/2 with values into the circle. C. R. Acad. Sci., Paris, Sér. I, Math. 331, 119–124 (2000) Bredon, G.E.: Topology and geometry. Corrected third printing of the 1993 original. Graduate Texts in Mathematics, 139. New York: Springer 1997 Brezis, H., Coron, J.-M., Lieb, E.: Harmonic maps with defects. Commun. Math. Phys. 107, 649–705 (1986) Demengel, F.: Une caractérisation des applications de W 1,1(B n,S 1) qui peuvent être approchées par des fonctions régulières [A characterization of maps in W 1,1(B n,S 1) that can be approximated by smooth maps]. C. R. Acad. Sci., Paris, Sér. I, Math. 310, 553–557 (1990) Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions. Studies in Advanced Mathematics. Boca Raton: CRC Press 1992 Federer, H.: Geometric measure theory. Grundlehren der mathematischen Wissenschaften, 153. New York: Springer 1969. Reprinted in the series Classics in Mathematics. Berlin, Heidelberg: Springer 1996 Gagliardo, E.: Caratterizzazione delle traccie sulla frontiera relativa ad alcune classi di funzioni in piú variabili. Rend. Semin. Mat. Univ Padova 27, 284–305 (1957) Giaquinta, M., Modica, G., Souček, J.: Cartesian currents, weak diffeomorphism, and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 106, 97–159 (1989). Erratum and addendum, Arch. Ration. Mech. Anal. 109, 385–392 (1990) Giaquinta, M., Modica, G., Souček, J.: Cartesian currents in the calculus of variations. I. Cartesian currents. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 37. Berlin: Springer 1998 Giaquinta, M., Modica, G., Souček, J.: Cartesian currents in the calculus of variations. II. Variational integrals. Ergeb. Math. Grenzgeb., 3. Folge, 38. Berlin: Springer 1998 Guillemin, V., Pollack, A.: Differential topology. Englewood Cliffs, N.J.: Prentice-Hall 1974 Hang, F.-B., Lin, F.-H.: A remark on the Jacobians. Commun. Contemp. Math. 2, 35–46 (2000) Hardt, R., Rivière, T.: Connecting topological Hopf singularities. To appear in Ann. Scuola Norm. Sup. Pisa Hirsch, M.W.: Differential topology. Corrected reprint of the 1976 original. Graduate Texts in Mathematics, 33. New York: Springer 1994 Jerrard, R.L.: A new proof of the rectifiable slices theorem. Ann. Scuola Norm. Sup. Pisa (5) 1, 905–924 (2002) Jerrard, R.L., Soner, H.M.: Rectifiability of the distributional Jacobian for a class of functions. C. R. Acad. Sci., Paris, Sér. I, Math. 329, 683–688 (1999) Jerrard, R.L., Soner, H.M.: Functions of bounded higher variation. Indiana Univ. Math. J. 51, 645–677 (2002) Jerrard, R.L., Soner, H.M.: The Jacobian and the Ginzburg-Landau energy. Calc. Var. Partial Differ. Equ. 14, 151–191 (2002) Malý, J., Swanson, D., Ziemer, W.P.: The coarea formula for Sobolev mappings. Trans. Am. Math. Soc. 355, 477–492 (2003) Müller, S.: Det=det. A remark on the distributional determinant. C. R. Acad. Sci., Paris, Sér. I, Math. 311, 13–17 (1990) Pakzad, M.R.: On topological singular set of maps with finite 3-energy into S 3. Z. Anal. Anwend. 21, 529–530 (2002) Pakzad, M.R., Rivière, T.: Weak density of smooth maps for the Dirichlet energy between manifolds. Geom. Funct. Anal. 13, 223–257 (2003) Rivière, T.: Dense subsets of H 1/2(S 2,S 1). Ann. Global Anal. Geom. 18, 517–528 (2000) Simon, L.: Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis 3. Canberra: Australian National University 1983 Steenrod, N.: The topology of fibre bundles. Princeton Mathematical Series vol. 14. Princeton: Princeton University Press 1951. Reprinted in the series Princeton Landmarks in Mathematics. Princeton: Princeton University Press 1999 White, B.: Rectifiability of flat chains. Ann. Math. 150, 165–184 (1999)