Cartesian currents and variational problems for mappings into spheres
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 16 (1989) no. 3, pp. 393-485.
@article{ASNSP_1989_4_16_3_393_0,
     author = {Giaquinta, M. and Modica, G. and Sou\v{c}ek, J.},
     title = {Cartesian currents and variational problems for mappings into spheres},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     pages = {393--485},
     publisher = {Scuola normale superiore},
     volume = {Ser. 4, 16},
     number = {3},
     year = {1989},
     mrnumber = {1050333},
     zbl = {0713.49014},
     language = {en},
     url = {http://archive.numdam.org/item/ASNSP_1989_4_16_3_393_0/}
}
TY  - JOUR
AU  - Giaquinta, M.
AU  - Modica, G.
AU  - Souček, J.
TI  - Cartesian currents and variational problems for mappings into spheres
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 1989
SP  - 393
EP  - 485
VL  - 16
IS  - 3
PB  - Scuola normale superiore
UR  - http://archive.numdam.org/item/ASNSP_1989_4_16_3_393_0/
LA  - en
ID  - ASNSP_1989_4_16_3_393_0
ER  - 
%0 Journal Article
%A Giaquinta, M.
%A Modica, G.
%A Souček, J.
%T Cartesian currents and variational problems for mappings into spheres
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 1989
%P 393-485
%V 16
%N 3
%I Scuola normale superiore
%U http://archive.numdam.org/item/ASNSP_1989_4_16_3_393_0/
%G en
%F ASNSP_1989_4_16_3_393_0
Giaquinta, M.; Modica, G.; Souček, J. Cartesian currents and variational problems for mappings into spheres. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 16 (1989) no. 3, pp. 393-485. http://archive.numdam.org/item/ASNSP_1989_4_16_3_393_0/

[1] F. Almgrem - W. Browder - E.H. Lieb, Co-area, Liquid crystals, and minimal surfaces, In DDT - a selection of Papers, Springer-Verlag, 1987. | MR

[2] F. Almgrem - E.H. Lieb, Singularities of energy minimizing maps from the ball to the sphere: examples, counterexamples, and bounds, Preprint.

[3] L. Ambrosio - S. Mortola - V.M. Tortorelli, Functionals with linear growth defined on vector valued BV functions, preprint. | MR

[4] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal. 63 (1977). | MR | Zbl

[5] J.M. Ball, Global invertibility of Sobolev functions and the interpenetration of matter, Proc. Roy. Soc. Edinburgh A 88 (1981) 315-328. | MR | Zbl

[6] J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos. Trans. Roy. Soc. A 306 (1982) 577-611. | MR | Zbl

[7] J.M. Ball - F. Murat, W1.p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984) 225-253. | MR | Zbl

[8] F. Bethuel, A characterization of maps in H1 (B3,S 2) which can be approximated by smooth maps, preprint.

[9] F. Bethuel, Approximation dans des espaces de Sobolev entre deux variété et groupes d' homotopie, preprint. | MR

[10] F. Bethuel - X. Zheng, Density of smooth functions between two manifolds in Sobolev spaces, preprint. | MR

[11] J. Blat - Jh. Morel, Elliptic problem image segmentation and their relation to fracture theory, preprint.

[12] R. Bott - L.W. Tu, Differential forms in Algebraic Topology, Springer-Verlag, New York, 1982. | MR | Zbl

[13] S. Brezis, Metastable harmonic maps, In "Metastability and incompletely posed problems" Eds. S. Antman, J.L. Ericksen, D. Kinderlehrer, J. Müller, Springer-Verlag 1987. | MR

[14] S. Brezis, Sk-valued maps with singularities, In "Topics in Calculus of Variations" Ed. M. Giaquinta, Lecture Notes in Math. n.1365, Springer-Verlag 1989. | MR | Zbl

[15] S. Brezis - J.M. Coron, Large solutions for harmonic maps in two dimensions, Commun. Math. Phys. 92 (1983) 203-215. | MR | Zbl

[16] S. Brezis - J.M. Coron - E.H. Lieb, Harmonic maps with defects, Comm. Math. Phys. 107 (1986), 649-705. | MR | Zbl

[17] H. Buseman G. Ewald - G. Shepard, Convex bodies and convexity on Grassmann cones, Part I to IV Math. Ann. 151 (1963) 1-41. | MR | Zbl

[18] P.J. Ciarlet - J. Ne, Unilateral problems in nonlinear, three-dimensional elasticity, Arch. Rat. Mech. Anal. 97 (1987) 171-188.

[19] B. Dacorogna, Remarques sur les notions de polyconvexité, quasi-convexité et convexité de rang 1. J. Math. pures et Appl. 64 (1985) 403-438. | MR | Zbl

[20] G. Dal Maso, Integral representation on BV(Ω) of r-limits of variational integrals. Manuscripta Math. 30 (1980) 387-416. | Zbl

[21] E. De Giorgi - L. Ambrosio, Un nuovo tipo di funzionale del calcolo delle variazioni, preprint.

[22] E. De Giorgi - M. Carriero - A. Leaci, Existence theorem for a minimum problem with free discontinuity set, preprint.

[23] J. Eells - L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978) 1-69. | MR | Zbl

[24] J. Eells - L. Lemaire, Selected topics in harmonic maps, Amer. Mat. Soc. Regional Conf. Series in Math. 50, 1983. | MR | Zbl

[25] J. Eells - L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988) 385-524. | MR | Zbl

[26] J. Eells - J.B. Wood, Restrictions on harmonic maps of surfaces, Topology 15 (1976) 263-266. | MR | Zbl

[27] I. Ekeland - R. Temam, Convex analysis and variational problems, North Holland, Amsterdam 1976. | MR | Zbl

[28] J. Ericksen - D. Kinderlehrer, Theory and applications of liquid crystals, IMA Series vol 5, Springer-Verlag 1987. | MR | Zbl

[29] M.J. Esteban, A direct variational approach to Skyrme's model for meson fields, Comm. Math. Phys. 105 (1986) 571-591. | MR | Zbl

[30] H. Federer, Geometric measure theory, Springer-Verlag, New York, 1969. | MR | Zbl

[31] H. Federer - W. Fleming, Normal and integral currents, Ann. of Math. 72 (1960) 458-520. | MR | Zbl

[32] M. Giaquinta - E. Giusti, The singular set of the minima of certain quadratic functionals, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), 45-55. | Numdam | MR | Zbl

[33] M. Giaquinta - G. Modica - J. Sou, Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity, Archive for Rat. Mech. Anal. 106 (1989) 97-159. Erratum and addendum, to appear in Archive for Rat. Mech. Anal. | MR | Zbl

[34] R. Hardt - D. Kinderlehrer - F.H. Lin, Existence and partial regularity of static liquid crystal configurations, Comm. Math. Phys. 105 (1986), 547-570. | MR | Zbl

[35] R. Hardt - D. Kinderlehrer - F.H. Lin, A remark about the stability of smooth equilibrium configurations of static liquid crystals, Mol. Cryst. Liq, Cryst. 139 (1986) 189-194.

[36] R. Hardt - D. Kinderlehrer - F.H. Lin, Stable defects of minimizers of constrained variational principles, Ann. Inst. Henri Poincaré 5 (1988) 207-322. | Numdam | MR | Zbl

[37] R. Hardt - F.H. Lin, A remark on H1 mappings, Manuscripta Math. 56 (1986) 1010. | MR | Zbl

[38] R. Hardt - L. Simon, Seminar on Geometric Measure Theory, Birkhäuser, Boston, 1986. | MR | Zbl

[39] J. Jost, The Dirichlet problem for harmonic maps from a surface with boundary into a 2-sphere with non-constant boundary values, J. Diff. Geom. 19 (1984) 393-401. | MR | Zbl

[40] J. Jost, Harmonic maps between surfaces, Lecture notes in Math. 1062, Springer-Verlag 1984. | MR | Zbl

[41] R.V. Kohn - G. Strang, Optimal design and relaxation of variational problems, Comm. Pure Appl. Math. to appear.

[42] L. Lemaire, Applications harmoniques de surfaces riemanniennes, J.Diff. Geom. 13 (1978) 51-78. | MR | Zbl

[43] P.L. Lions, The concentration compactness principle in the calculus of variations. The limit case, part 2, Revista Matematica Iberoamericana 1 (1985) 45-121. | MR | Zbl

[44] F.C. Liu, A Lusin type property of Sobolev functions, Indiana Univ. Math. J. 26 (1977) 645-671. | MR | Zbl

[45] P. Marcellini, The stored energy of some discontinuous deformations in nonlinear elasticity, In Partial Differential Equations and the Calculus of Variations: Essays in Honor of Ennio De Giorgi, Birkäuser, Boston, 1989, to appear. | MR | Zbl

[46] M. Meier, Removable singularities of harmonic maps and an application to minimized submanifolds, Indiana Univ. Math. J. 35 (1986) 705-726. | MR | Zbl

[47] J.M. Morel, S. Solimini, Segmentation of images by variational methods, preprint.

[48] F. Morgan, Geometric measure theory, Academic Press, 1988. | MR | Zbl

[49] C.B. Morrey, Multiple integrals in the calculus of variations, Springer-Verlag, Berlin, 1966. | MR | Zbl

[50] S. Muller, Weak continuity of determinants and nonlinear elasticity, C.R. Acad. Sci. Paris 307 (1988) 501-576. | MR | Zbl

[51] D. Mumford, J. Shah, Boundary detection by minimizing functionals, Proceedings of the IEEE. Conference on computer vision and pattern recognition, San Francisco, 1985.

[52] L. Nirenberg, Topics in Nonlinear Functionals Analysis, New York University, Lectures Notes, New York (1974). | MR | Zbl

[53] T. Qi, Almost-everywhere injectivity in nonlinear elasticity, Proc. Roy. Soc. Edinburgh 109A (1988) 79-95. | MR | Zbl

[54] Y.G. Reshetnyak, On the stability of conformal mappings in multidimensional spaces, Siberian Math. J. 8 (1967) 69-85. | Zbl

[55] Y.G. Reshetnyak, Stability theorems for mappings with bounded distorsion, Siberian Math. J. 9 (1968) 499-512. | MR | Zbl

[56] Y.G. Reshetnyak, Weak convergence of completely additive vector functions on a set, Sibirskii Mat. Zhurnal 9 (1968) 1386-1394. | MR | Zbl

[57] R. Schoen - K. Uhlenbeck, A regularity theory for harmonic maps, J. Diff. Geom. 17 (1982) 307-335. | MR | Zbl

[58] R. Schoen - K. Uhlenbeck, Boundary regularity and miscellaneous results on harmonic maps, J. Diff. Geom. 18 (1983) 253-268. | MR | Zbl

[59] L. Simon, Lectures on Geometric Measure Theory, Proc. of the Centre for Math. Analysis vol. 3 Australian National University, Canberra, 1983. | MR | Zbl

[60] A. Soyeur, The Dirichlet problem for harmonic maps from the disc into 2-sphere, preprint. | MR

[61] V. Šverák, Regularity properties of deformations with finite energy, Arch. Rat. Mech. Anal. 98 (1987). | MR | Zbl

[62] B. White, A new proof of the compactness theorem for integral currents, preprint. | MR