Cartesian currents and variational problems for mappings into spheres

 Access to full text

 Full (PDF)

1. [1] F. Almgrem - W. Browder - E.H. Lieb, Co-area, Liquid crystals, and minimal surfaces, In DDT - a selection of Papers, Springer-Verlag, 1987. Zbl0645.58015MR1032767
2. [2] F. Almgrem - E.H. Lieb, Singularities of energy minimizing maps from the ball to the sphere: examples, counterexamples, and bounds, Preprint. 
3. [3] L. Ambrosio - S. Mortola - V.M. Tortorelli, Functionals with linear growth defined on vector valued BV functions, preprint. Zbl0662.49007MR1113814
4. [4] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal.63 (1977). Zbl0368.73040MR475169
5. [5] J.M. Ball, Global invertibility of Sobolev functions and the interpenetration of matter, Proc. Roy. Soc. EdinburghA88 (1981) 315-328. Zbl0478.46032MR616782
6. [6] J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Philos. Trans. Roy. Soc. A306 (1982) 577-611. Zbl0513.73020MR703623
7. [7] J.M. Ball - F. Murat, W1.p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal.58 (1984) 225-253. Zbl0549.46019MR759098
8. [8] F. Bethuel, A characterization of maps in H1 (B3,S 2) which can be approximated by smooth maps, preprint. Zbl0708.58004
9. [9] F. Bethuel, Approximation dans des espaces de Sobolev entre deux variété et groupes d' homotopie, preprint. MR958784
10. [10] F. Bethuel - X. Zheng, Density of smooth functions between two manifolds in Sobolev spaces, preprint. MR960223
11. [11] J. Blat - JH. Morel, Elliptic problem image segmentation and their relation to fracture theory, preprint. Zbl0727.35040
12. [12] R. Bott - L.W. Tu, Differential forms in Algebraic Topology, Springer-Verlag, New York, 1982. Zbl0496.55001MR658304
13. [13] S. Brezis, Metastable harmonic maps, In "Metastability and incompletely posed problems" Eds. S. Antman, J.L. Ericksen, D. Kinderlehrer, J. Müller, Springer-Verlag1987. MR870008
14. [14] S. Brezis, Sk-valued maps with singularities, In "Topics in Calculus of Variations" Ed. M. Giaquinta, Lecture Notes in Math. n.1365, Springer-Verlag1989. Zbl0684.49015MR994017
15. [15] S. Brezis - J.M. Coron, Large solutions for harmonic maps in two dimensions, Commun. Math. Phys.92 (1983) 203-215. Zbl0532.58006MR728866
16. [16] S. Brezis - J.M. Coron - E.H. Lieb, Harmonic maps with defects, Comm. Math. Phys.107 (1986), 649-705. Zbl0608.58016MR868739
17. [17] H. Buseman G. Ewald - G. Shepard, Convex bodies and convexity on Grassmann cones, Part I to IVMath. Ann.151 (1963) 1-41. Zbl0112.37301MR157286
18. [18] P.J. Ciarlet - J. Ne, Unilateral problems in nonlinear, three-dimensional elasticity, Arch. Rat. Mech. Anal.97 (1987) 171-188. 
19. [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. Zbl0609.49007MR839729
20. [20] G. Dal Maso, Integral representation on BV(Ω) of r-limits of variational integrals. Manuscripta Math.30 (1980) 387-416. Zbl0435.49016
21. [21] E. De Giorgi - L. Ambrosio, Un nuovo tipo di funzionale del calcolo delle variazioni, preprint. 
22. [22] E. De Giorgi - M. Carriero - A. Leaci, Existence theorem for a minimum problem with free discontinuity set, preprint. Zbl0682.49002
23. [23] J. Eells - L. Lemaire, A report on harmonic maps, Bull. London Math. Soc.10 (1978) 1-69. Zbl0401.58003MR495450
24. [24] J. Eells - L. Lemaire, Selected topics in harmonic maps, Amer. Mat. Soc. Regional Conf. Series in Math.50, 1983. Zbl0515.58011MR703510
25. [25] J. Eells - L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc.20 (1988) 385-524. Zbl0669.58009MR956352
26. [26] J. Eells - J.B. Wood, Restrictions on harmonic maps of surfaces, Topology15 (1976) 263-266. Zbl0328.58008MR420708
27. [27] I. Ekeland - R. Temam, Convex analysis and variational problems, North Holland, Amsterdam1976. Zbl0322.90046MR463994
28. [28] J. Ericksen - D. Kinderlehrer, Theory and applications of liquid crystals, IMA Series vol 5, Springer-Verlag1987. Zbl0713.76006MR900827
29. [29] M.J. Esteban, A direct variational approach to Skyrme's model for meson fields, Comm. Math. Phys.105 (1986) 571-591. Zbl0621.58035MR852091
30. [30] H. Federer, Geometric measure theory, Springer-Verlag, New York, 1969. Zbl0176.00801MR257325
31. [31] H. Federer - W. Fleming, Normal and integral currents, Ann. of Math.72 (1960) 458-520. Zbl0187.31301MR123260
32. [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. Zbl0543.49018MR752579
33. [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. Zbl0677.73014MR980756
34. [34] R. Hardt - D. Kinderlehrer - F.H. Lin, Existence and partial regularity of static liquid crystal configurations, Comm. Math. Phys.105 (1986), 547-570. Zbl0611.35077MR852090
35. [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. [36] R. Hardt - D. Kinderlehrer - F.H. Lin, Stable defects of minimizers of constrained variational principles, Ann. Inst. Henri Poincaré5 (1988) 207-322. Zbl0657.49018MR963102
37. [37] R. Hardt - F.H. Lin, A remark on H1 mappings, Manuscripta Math.56 (1986) 1010. Zbl0618.58015MR846982
38. [38] R. Hardt - L. Simon, Seminar on Geometric Measure Theory, Birkhäuser, Boston, 1986. Zbl0601.49029MR891187
39. [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. Zbl0551.58012MR755231
40. [40] J. Jost, Harmonic maps between surfaces, Lecture notes in Math. 1062, Springer-Verlag1984. Zbl0542.58002MR754769
41. [41] R.V. Kohn - G. Strang, Optimal design and relaxation of variational problems, Comm. Pure Appl. Math. to appear. Zbl0609.49008
42. [42] L. Lemaire, Applications harmoniques de surfaces riemanniennes, J.Diff. Geom.13 (1978) 51-78. Zbl0388.58003MR520601
43. [43] P.L. Lions, The concentration compactness principle in the calculus of variations. The limit case, part 2, Revista Matematica Iberoamericana1 (1985) 45-121. Zbl0704.49006MR850686
44. [44] F.C. Liu, A Lusin type property of Sobolev functions, Indiana Univ. Math. J.26 (1977) 645-671. Zbl0368.46036MR450488
45. [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. Zbl0679.73006MR1034028
46. [46] M. Meier, Removable singularities of harmonic maps and an application to minimized submanifolds, Indiana Univ. Math. J.35 (1986) 705-726. Zbl0618.58017MR865424
47. [47] J.M. Morel, S. Solimini, Segmentation of images by variational methods, preprint. Zbl0679.68205
48. [48] F. Morgan, Geometric measure theory, Academic Press, 1988. Zbl0671.49043MR933756
49. [49] C.B. Morrey, Multiple integrals in the calculus of variations, Springer-Verlag, Berlin, 1966. Zbl0142.38701MR202511
50. [50] S. Muller, Weak continuity of determinants and nonlinear elasticity, C.R. Acad. Sci. Paris307 (1988) 501-576. Zbl0679.34051MR964116
51. [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. [52] L. Nirenberg, Topics in Nonlinear Functionals Analysis, New York University, Lectures Notes, New York (1974). Zbl0286.47037MR488102
53. [53] T. Qi, Almost-everywhere injectivity in nonlinear elasticity, Proc. Roy. Soc. Edinburgh109A (1988) 79-95. Zbl0656.73010MR952330
54. [54] Y.G. Reshetnyak, On the stability of conformal mappings in multidimensional spaces, Siberian Math. J.8 (1967) 69-85. Zbl0172.37801
55. [55] Y.G. Reshetnyak, Stability theorems for mappings with bounded distorsion, Siberian Math. J.9 (1968) 499-512. Zbl0176.03503MR230901
56. [56] Y.G. Reshetnyak, Weak convergence of completely additive vector functions on a set, Sibirskii Mat. Zhurnal9 (1968) 1386-1394. Zbl0176.44402MR240274
57. [57] R. Schoen - K. Uhlenbeck, A regularity theory for harmonic maps, J. Diff. Geom.17 (1982) 307-335. Zbl0521.58021MR664498
58. [58] R. Schoen - K. Uhlenbeck, Boundary regularity and miscellaneous results on harmonic maps, J. Diff. Geom.18 (1983) 253-268. Zbl0547.58020MR710054
59. [59] L. Simon, Lectures on Geometric Measure Theory, Proc. of the Centre for Math. Analysis vol. 3Australian National University, Canberra, 1983. Zbl0546.49019MR756417
60. [60] A. Soyeur, The Dirichlet problem for harmonic maps from the disc into 2-sphere, preprint. Zbl0701.58020MR1037728
61. [61] V. Šverák, Regularity properties of deformations with finite energy, Arch. Rat. Mech. Anal.98 (1987). Zbl0659.73038MR913960
62. [62] B. White, A new proof of the compactness theorem for integral currents, preprint. Zbl0706.49028MR997362

1. Mariano Giaquinta, Giuseppe Modica, Jiří Souček, Connectivity properties of the range of a weak diffeomorphism
2. Jan Malý, $L^p$-approximation of Jacobians
3. M. Giaquinta, G. Modica, J. Souček, Liquid crystals : relaxed energies, dipoles, singular lines and singular points
4. Paolo Maria Mariano, Giuseppe Modica, Ground states in complex bodies
5. M. Carriero, A. Leaci, $S^k$-valued maps minimizing the $L^p$ norm of the gradient with free discontinuities
6. Mariano Giaquinta, Giuseppe Modica, Jiří Souček, Some remarks about the $p$-Dirichlet integral
7. Paolo Maria Mariano, Giuseppe Modica, Ground states in complex bodies
8. Piotr Hajłasz, A note on weak approximation of minors
9. Irene Fonseca, Nicola Fusco, Paolo Marcellini, Topological degree, Jacobian determinants and relaxation