On a new class of elastic deformations not allowing for cavitation

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1. [Ad75] R. Adams, Sobolev Spaces, Academic Press, 1975. Zbl0314.46030MR450957
2. [Ba77] J.M. Ball, Convexity Conditions and Existence Theorems in Nonlinear Elasticity, Arch. Rat. Mech. Anal., Vol. 63, 1977, pp. 337-403. Zbl0368.73040MR475169
3. [Ba81] J.M. Ball, Global Invertibility of Sobolev Functions and the Interpenetration of Matter, Proc. Roy. Soc. Edinburgh, Vol. 88A, 1981, pp. 315-328. Zbl0478.46032MR616782
4. [Ba82] J.M. Ball, Discontinuous Equilibrium Solutions and Cavitation in Non-Linear Elasticity, Phil. Trans. Roy. Soc. London, Vol. 306A, 1982, pp. 557-612. Zbl0513.73020MR703623
5. [BM84] J.M. Ball and F. Murat, W1, p-Quasiconvexity and Variational Problems for Multiple Integrals, J. Funct. Anal., Vol. 58, 1984, pp. 225-253. Zbl0549.46019MR759098
6. [Be50] A.S. Besicovitch, Parametric Surfaces, Bull. Am. Math. Soc., Vol. 56, 1950, pp. 228-296. Zbl0038.20401MR36825
7. [BI83] B. Bojarski and T. Iwaniec, Analytical Foundations of the Theory of Quasiconformal Mappings in Rn, Ann. Acad. Sci. Fenn., Ser. A, Vol. 8, 1983, pp. 257-324. Zbl0548.30016MR731786
8. [BFS92] H. Brezis, N. Fusco and C. Sbordone, Integrability of the Jacobian of Orientation Preserving Mappings, J. Funct. Anal., Vol. 115, 1993, pp. 425-431. Zbl0847.26012MR1234399
9. [CN87] P.G. Ciarlet and J. Necas, Injectivity and Self-Contact in Non-Linear Elasticity, Arch. Rat. Mech. Anal., Vol. 97, 1987, pp. 171-188. Zbl0628.73043MR862546
10. [CLMS89] R. Coifman, P.L. Lions, Y. Meyer and S. Semmes, Compacité par compensation et espaces de Hardy, C.R. Acad. Sci. Paris, T. 309, Serie I, 1989, pp. 945-949; Compensated Compactness and Hardy Spaces, J. Math Pures Appl., Vol. 72, 1993, pp. 247-286. Zbl0864.42009
11. [Da89] B. Dacorogna, Direct Methods in the Calculus of Variations, Springer, 1989. Zbl0703.49001MR990890
12. [EG91] L.C. Evans and R.E. Gariepy, Lecture Notes on Measure Theory and Fine Properties of Functions, C.R.C. Publ., 1991. 
13. [Fe69] H. Federer, Geometric Measure Theory, Springer, 1969. Zbl0176.00801MR257325
14. [F163] H. Flanders, Differential Forms, Academic Press, 1963. Zbl0112.32003MR162198
15. [GT83] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2nd ed., 1983. Zbl0562.35001MR737190
16. [Tu84] E. Giusti, Minimal Surfaces and Functions of Bounded Variations, Birkhäuser, 1984. Zbl0545.49018MR775682
17. [GI92] L. Greco and T. Iwaniec, New Inequalities for the Jacobian, Preprint. 
18. [IL92] T. Iwaniec and A. Lutoborski, Integral Estimatesfor Null Lagrangians, Preprint, Syracuse University. MR1241286
19. [IS91] T. Iwaniec and C. Sbordone, On the Integrability of the Jacobian Under Minimal Hypotheses, Arch. Rat. Mech. Anal., Vol. 119, 1992, pp. 129-143. Zbl0766.46016MR1176362
20. [MM92] J. Maly and O. Martio, Lusin's Condition (N) and Mappings of the Class W1, n, Preprint. Zbl0812.30007
21. [Ma92] J.J. Manfredi, Weakly Monotone Functions, Preprint. MR1294334
22. [Me66] P.A. Meyer, Probability and Potentials, Waltham, 1966. Zbl0138.10401MR205288
23. [MM73] M. Marcus and V.J. Mizel, Transformations by Functions in Sobolev Spaces and Lower Semicontinuity for Parametric Variational Problems, Bull. Amer Math. Soc., Vol. 79, 1973, pp. 790-795. Zbl0275.49041MR322651
24. [Mo66] C.B. Morrey, Multiple Integrals in the Calculus of Variations, Springer, 1966. Zbl0142.38701
25. [MS47] E.J. Mcshane, Integration, Princeton Univ. press, 1947. Zbl0033.05302MR82536
26. [Mu89] S. Müller, A Surprising Higher Integrability Property of Mappings with Positive Determinant, Bull. Amer Math. Soc., Vol. 21, 1989, pp. 245-248. Zbl0689.49006MR999618
27. [Mu90 a] S. Müller, Det = det. A Remark on the Distributional Determinant, C.R. Acad. Sci., Paris, Vol. 311, 1990, pp. 13-17. Zbl0717.46033MR1062920
28. [Mu90 b] S. Müller, Higher Integrability of Determinants and Weak Convergence in L1, J. reine angew. Math., Vol. 412, 1990, pp. 20-34. Zbl0713.49004MR1078998
29. [Mu91 a] S. Müller, A Counter-Example Concerning Formal Integration by Parts, C.R. Acad. Sci., Paris, Vol. 312, 1991, pp. 45-49. Zbl0723.46028
30. [Mu91 b] S. Müller, On the Singular Support of the Distributional Determinant, Ann. Inst. H. Poincaré, Analyse non linéaire, Vol. 10, 1993, pp. 657-696. Zbl0792.46027MR1253606
31. [MS92] S. Müller and S.J. Spector, Existence Theorems in Nonlinear Elasticity Allowing for Cavitation, submitted to Arch. Rat. Mech. Anal. 
32. [MST91] S. Müller, S.J. Spector and Q. Tang, Invertibility and a Topological Property of Sobolev Maps (in preparation). Zbl0855.73028
33. [Ne67] J. Necas, Les méthodes directes en théorie des équations elliptiques, Masson, 1967. MR227584
34. [Og72] R.W. Ogden, Large Deformation Isotropic Elasticity — On the Correlation of Theory and Experiment for Incompressible Rubber-Like Solids, Proc. Roy. Soc. London, Vol. A326, 1972, pp. 565-584. Zbl0257.73034
35. [Re67] Y.G. Reshetnyak, On the Stability of Conformal Mappings in Multidimensional Spaces, Siberian Math. J., Vol. 8, 1967, pp. 65-85. Zbl0172.37801
36. [Re89] Y.G. Reshetnyak, Space Mappings with Bounded Distorsion, Transl. Math. Monographs, Vol. 73, Ann. Math. Soc., 1989. Zbl0667.30018MR994644
37. [Ri93] S. Rickman, Quasiregular Mappings, Springer, 1993. Zbl0816.30017MR1238941
38. [Sc69] J.T. Schwartz, Nonlinear Functional Analysis, Acad. Press, 1969. Zbl0203.14501MR433481
39. [Si83] L. Simon, Lectures on Geometric Measure Theory, Centre Math. Anal. Australian National University, 1983. Zbl0546.49019MR756417
40. [Sv88] V. Šverák, Regularity Properties of Deformations with Finite Energy, Arch. Rat. Mech. Anal., Vol. 100, 1988, pp. 105-127. Zbl0659.73038MR913960
41. [TQ88] Q. Tang, Almost-Everywhere Injectivity in Nonlinear Elasticity, Proc. Roy. Soc. Edinburgh, Vol. 109A, 1988, pp. 79-95. Zbl0656.73010MR952330
42. [VG77] S.K. Vodopyanov and V.M. Goldstein, Quasiconformal Mappings and Spaces of Functions with Generalised First Derivatives, Siberian Math. J., Vol. 12, 1977, pp. 515-531. 
43. [Zh90] K.W. Zhang, Biting Theorems for Jacobians and their Applications, Ann. Inst. H. Poincaré, Analyse non linéaire, Vol. 7, 1990, pp. 345-365. Zbl0717.49012MR1067780

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