Weak notions of jacobian determinant and relaxation

 Access to full text

1. [1] G. Alberti, S. Baldo and G. Orlandi, Functions with prescribed singularities. J. Eur. Math. Soc. (JEMS) 5 (2003) 275–311. Zbl1033.46028MR2002215
2. [2] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (2000). Zbl0957.49001MR1857292
3. [3] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal.63 (1976) 337–403. Zbl0368.73040MR475169
4. [4] F. Bethuel, A characterization of maps in H1(B3,S2) which can be approximated by smooth maps. Ann. Inst. Henri Poincaré Anal. Non Linéaire7 (1990) 269–286. Zbl0708.58004MR1067776
5. [5] F. Bethuel, The approximation problem for Sobolev maps between two manifolds. Acta Math.167 (1991) 153–206. Zbl0756.46017MR1120602
6. [6] G. Bouchitté, I. Fonseca and J. Malý, The effective bulk energy of the relaxed energy of multiple integrals below the growth exponent. Proc. R. Soc. Edinb. Sect. A128 (1998) 463–479. Zbl0907.49008MR1632814
7. [7] H. Brezis and L. Nirenberg, Degree theory and BMO. I. Compact manifolds without boundaries. Selecta Mathematica (N.S.) 1 (1995) 197–263. Zbl0852.58010MR1354598
8. [8] H. Brezis and L. Nirenberg, Degree theory and BMO. II. Compact manifolds with boundaries. Selecta Mathematica (N.S.) 2 (1996) 309–368. Zbl0868.58017MR1422201
9. [9] H. Brezis, J.-M. Coron and E.H. Lieb, Harmonic maps with defects. Comm. Math. Phys.107 (1986) 649–705. Zbl0608.58016MR868739
10. [10] R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces. J. Math. Pures Appl. (9) 72 (1993) 247–286. Zbl0864.42009MR1225511
11. [11] S. Conti and C. De Lellis, Some remarks on the theory of elasticity for compressible Neohookean materials. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 2 (2003) 521–549. Zbl1114.74004MR2020859
12. [12] B. Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences 78. Springer, New York, second edition (2008). Zbl1140.49001MR2361288
13. [13] B. Dacorogna and P. Marcellini, Semicontinuité pour des intégrandes polyconvexes sans continuité des déterminants. C. R. Acad. Sci. Paris Sér. I Math.311 (1990) 393–396. Zbl0723.49007MR1071650
14. [14] C. De Lellis, Some fine properties of currents and applications to distributional Jacobians. Proc. R. Soc. Edinb. Sect. A132 (2002) 815–842. Zbl1025.49029MR1926918
15. [15] C. De Lellis, Some remarks on the distributional Jacobian. Nonlinear Anal.53 (2003) 1101–1114. Zbl1025.49030MR1978037
16. [16] C. De Lellis and F. Ghiraldin, An extension of Müller’s identity Det = det. C. R. Math. Acad. Sci. Paris348 (2010) 973–976. Zbl1201.35088
17. [17] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992). Zbl0804.28001MR1158660
18. [18] H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York (1969). Zbl0176.00801MR257325
19. [19] I. Fonseca and W. Gangbo, Degree theory in analysis and applications, Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press Oxford University Press, New York (1995). Zbl0852.47030MR1373430
20. [20] I. Fonseca and J. Malý, Relaxation of multiple integrals below the growth exponent. Ann. Inst. Henri Poincaré Anal. Non Linéaire14 (1997) 309–338. Zbl0868.49011MR1450951
21. [21] I. Fonseca and P. Marcellini, Relaxation of multiple integrals in subcritical Sobolev spaces. J. Geom. Anal.7 (1997) 57–81. Zbl0915.49011MR1630777
22. [22] I. Fonseca, N. Fusco and P. Marcellini, On the total variation of the Jacobian. J. Funct. Anal.207 (2004) 1–32. Zbl1041.49016MR2027634
23. [23] I. Fonseca, N. Fusco and P. Marcellini, Topological degree, Jacobian determinants and relaxation. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 8 (2005) 187–250. Zbl1177.49066MR2122983
24. [24] M. Giaquinta, G. Modica and J. Souček, Graphs of finite mass which cannot be approximated in area by smooth graphs. Manuscr. Math.78 (1993) 259–271. Zbl0796.58006MR1206156
25. [25] M. Giaquinta, G. Modica and J. Souček, Remarks on the degree theory. J. Funct. Anal.125 (1994) 172–200. Zbl0822.55003MR1297018
26. [26] M. Giaquinta, G. Modica and J. Souček, Cartesian currents in the calculus of variations I, Cartesian currents. Springer-Verlag, Berlin (1998). Zbl0914.49001MR1645086
27. [27] A. Hatcher, Algebraic topology. Cambridge University Press, Cambridge (2002). Zbl1044.55001MR1867354
28. [28] R.L. Jerrard and H.M. Soner, Functions of bounded higher variation. Indiana Univ. Math. J.51 (2002) 645–677. Zbl1057.49036MR1911049
29. [29] J. Malý, Lp-approximation of Jacobians. Comment. Math. Univ. Carolin.32 (1991) 659–666. Zbl0753.46024MR1159812
30. [30] P. Marcellini, Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals. Manuscr. Math.51 (1985) 1–28. Zbl0573.49010MR788671
31. [31] P. Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals. Ann. Inst. Henri Poincaré Anal. Non Linéaire3 (1986) 391–409. Zbl0609.49009MR868523
32. [32] P. Marcellini, The stored-energy for some discontinuous deformations in nonlinear elasticity, in Partial differential equations and the calculus of variations II, Progr. Nonlinear Differential Equations Appl. 2, Birkhäuser Boston, Boston, MA (1989) 767–786. Zbl0679.73006MR1034028
33. [33] D. Mucci, Remarks on the total variation of the Jacobian. NoDEA Nonlinear Differential Equations Appl.13 (2006) 223–233. Zbl1117.49017MR2243712
34. [34] D. Mucci, A variational problem involving the distributional determinant. Riv. Mat. Univ. Parma (to appear). Zbl1222.49056MR2789447
35. [35] S. Müller, Higher integrability of determinants and weak convergence in L1. J. Reine Angew. Math.412 (1990) 20–34. Zbl0713.49004MR1078998
36. [36] S. Müller, Det = det. A remark on the distributional determinant. C. R. Acad. Sci. Paris Sér. I Math. 311 (1990) 13–17. Zbl0717.46033MR1062920
37. [37] S. Müller, On the singular support of the distributional determinant. Ann. Inst. Henri Poincaré Anal. Non Linéaire10 (1993) 657–696. Zbl0792.46027MR1253606
38. [38] S. Müller and S.J. Spector, An existence theory for nonlinear elasticity that allows for cavitation. Arch. Rational Mech. Anal.131 (1995) 1–66. Zbl0836.73025MR1346364
39. [39] S. Müller, Q. Tang and B.S. Yan, On a new class of elastic deformations not allowing for cavitation. Ann. Inst. Henri Poincaré Anal. Non Linéaire11 (1994) 217–243. Zbl0863.49002MR1267368
40. [40] S. Müller, S.J. Spector and Q. Tang, Invertibility and a topological property of Sobolev maps. SIAM J. Math. Anal.27 (1996) 959–976. Zbl0855.73028MR1393418
41. [41] E. Paolini, On the relaxed total variation of singular maps. Manuscr. Math.111 (2003) 499–512. Zbl1023.49010MR2002824
42. [42] A.C. Ponce and J. Van Schaftingen, Closure of smooth maps in W1, p(B3;S2). Differential Integral Equations22 (2009) 881–900. Zbl1240.46063MR2553061
43. [43] T. Schmidt, Regularity of Relaxed Minimizers of Quasiconvex Variational Integrals with (p, q)-growth. Arch. Rational Mech. Anal.193 (2009) 311–337. Zbl1173.49032MR2525120
44. [44] B. White, Existence of least-area mappings of N-dimensional domains. Ann. Math. (2) 118 (1983) 179–185. Zbl0526.49029MR707165

1. Luigi Ambrosio, Francesco Ghiraldin, Compactness of Special Functions of Bounded Higher Variation