On a new class of elastic deformations not allowing for cavitation
Annales de l'I.H.P. Analyse non linéaire, Volume 11 (1994) no. 2, pp. 217-243.
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author = {M\"uller, S. and Qi, Tang and Yan, B. S.},
title = {On a new class of elastic deformations not allowing for cavitation},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {217--243},
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Müller, S.; Qi, Tang; Yan, B. S. On a new class of elastic deformations not allowing for cavitation. Annales de l'I.H.P. Analyse non linéaire, Volume 11 (1994) no. 2, pp. 217-243. http://archive.numdam.org/item/AIHPC_1994__11_2_217_0/

[Ba77] J.M. Ball, Convexity Conditions and Existence Theorems in Nonlinear Elasticity, Arch. Rat. Mech. Anal., Vol. 63, 1977, pp. 337-403. | MR | Zbl

[Ba81] J.M. Ball, Global Invertibility of Sobolev Functions and the Interpenetration of Matter, Proc. Roy. Soc. Edinburgh, Vol. 88A, 1981, pp. 315-328. | MR | Zbl

[Ba82] J.M. Ball, Discontinuous Equilibrium Solutions and Cavitation in Non-Linear Elasticity, Phil. Trans. Roy. Soc. London, Vol. 306A, 1982, pp. 557-612. | MR | Zbl

[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. | MR | Zbl

[Be50] A.S. Besicovitch, Parametric Surfaces, Bull. Am. Math. Soc., Vol. 56, 1950, pp. 228-296. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | Zbl

[Da89] B. Dacorogna, Direct Methods in the Calculus of Variations, Springer, 1989. | MR | Zbl

[EG91] L.C. Evans and R.E. Gariepy, Lecture Notes on Measure Theory and Fine Properties of Functions, C.R.C. Publ., 1991.

[Fe69] H. Federer, Geometric Measure Theory, Springer, 1969. | MR | Zbl

[F163] H. Flanders, Differential Forms, Academic Press, 1963. | MR | Zbl

[GT83] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2nd ed., 1983. | MR | Zbl

[Tu84] E. Giusti, Minimal Surfaces and Functions of Bounded Variations, Birkhäuser, 1984. | MR | Zbl

[GI92] L. Greco and T. Iwaniec, New Inequalities for the Jacobian, Preprint.

[IL92] T. Iwaniec and A. Lutoborski, Integral Estimatesfor Null Lagrangians, Preprint, Syracuse University. | MR

[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. | MR | Zbl

[MM92] J. Maly and O. Martio, Lusin's Condition (N) and Mappings of the Class W1, n, Preprint.

[Ma92] J.J. Manfredi, Weakly Monotone Functions, Preprint. | MR

[Me66] P.A. Meyer, Probability and Potentials, Waltham, 1966. | MR | Zbl

[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. | MR | Zbl

[Mo66] C.B. Morrey, Multiple Integrals in the Calculus of Variations, Springer, 1966. | Zbl

[MS47] E.J. Mcshane, Integration, Princeton Univ. press, 1947. | MR | Zbl

[Mu89] S. Müller, A Surprising Higher Integrability Property of Mappings with Positive Determinant, Bull. Amer Math. Soc., Vol. 21, 1989, pp. 245-248. | MR | Zbl

[Mu90 a] S. Müller, Det = det. A Remark on the Distributional Determinant, C.R. Acad. Sci., Paris, Vol. 311, 1990, pp. 13-17. | MR | Zbl

[Mu90 b] S. Müller, Higher Integrability of Determinants and Weak Convergence in L1, J. reine angew. Math., Vol. 412, 1990, pp. 20-34. | MR | Zbl

[Mu91 a] S. Müller, A Counter-Example Concerning Formal Integration by Parts, C.R. Acad. Sci., Paris, Vol. 312, 1991, pp. 45-49. | Zbl

[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. | Numdam | MR | Zbl

[MS92] S. Müller and S.J. Spector, Existence Theorems in Nonlinear Elasticity Allowing for Cavitation, submitted to Arch. Rat. Mech. Anal.

[MST91] S. Müller, S.J. Spector and Q. Tang, Invertibility and a Topological Property of Sobolev Maps (in preparation).

[Ne67] J. Necas, Les méthodes directes en théorie des équations elliptiques, Masson, 1967. | MR

[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. | Zbl

[Re67] Y.G. Reshetnyak, On the Stability of Conformal Mappings in Multidimensional Spaces, Siberian Math. J., Vol. 8, 1967, pp. 65-85. | Zbl

[Re89] Y.G. Reshetnyak, Space Mappings with Bounded Distorsion, Transl. Math. Monographs, Vol. 73, Ann. Math. Soc., 1989. | MR | Zbl

[Ri93] S. Rickman, Quasiregular Mappings, Springer, 1993. | MR | Zbl

[Sc69] J.T. Schwartz, Nonlinear Functional Analysis, Acad. Press, 1969. | MR | Zbl

[Si83] L. Simon, Lectures on Geometric Measure Theory, Centre Math. Anal. Australian National University, 1983. | MR | Zbl

[Sv88] V. Šverák, Regularity Properties of Deformations with Finite Energy, Arch. Rat. Mech. Anal., Vol. 100, 1988, pp. 105-127. | MR | Zbl

[TQ88] Q. Tang, Almost-Everywhere Injectivity in Nonlinear Elasticity, Proc. Roy. Soc. Edinburgh, Vol. 109A, 1988, pp. 79-95. | MR | Zbl

[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.

[Zh90] K.W. Zhang, Biting Theorems for Jacobians and their Applications, Ann. Inst. H. Poincaré, Analyse non linéaire, Vol. 7, 1990, pp. 345-365. | Numdam | MR | Zbl