On a new class of elastic deformations not allowing for cavitation. (English) Zbl 0863.49002

The divergence identity \[ \partial_j\{(g^i\circ u)(\text{adj }Du)^j_i\}= (\text{div }g)\circ\text{det }Du \] is proved under weak assumptions. Also, an integrability result for \(\text{det }Du\) is proved. As an application, a degree formula is proved in the class \(A_{p,q}(\Omega)\). As a further application, the existence of an absolutely continuous minimizer of \[ I(u)=\int_\Omega W(x,u,Du)dx \] is proved if \(W\) is polyconvex and satisfies the inequality \(W(x,u,F)\geq a(|F|^p+|\text{adj }F|^q)\) with \(p\geq 2\) and \(q\geq 3/2\).


49J10 Existence theories for free problems in two or more independent variables
26B10 Implicit function theorems, Jacobians, transformations with several variables
74B20 Nonlinear elasticity
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[1] Adams, R., Sobolev spaces, (1975), Academic Press · Zbl 0314.46030
[2] Ball, J. M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal., Vol. 63, 337-403, (1977) · Zbl 0368.73040
[3] Ball, J. M., Global invertibility of Sobolev functions and the interpenetration of matter, Proc. Roy. Soc. Edinburgh, Vol. 88A, 315-328, (1981) · Zbl 0478.46032
[4] Ball, J. M., Discontinuous equilibrium solutions and cavitation in non-linear elasticity, Phil. Trans. Roy. Soc. London, Vol. 306A, 557-612, (1982) · Zbl 0513.73020
[5] Ball, J. M.; Murat, F., W^{1, p}-quasiconvexity and variational problems for multiple integrals, J. Fund. Anal., Vol. 58, 225-253, (1984) · Zbl 0549.46019
[6] Besicovitch, A. S., Parametric surfaces, Bull. Am. Math. Soc., Vol. 56, 228-296, (1950) · Zbl 0038.20401
[7] Bojarski, B.; Iwaniec, T., Analytical foundations of the theory of quasicon-formal mappings in ℝ^{n}, Ann. Acad. Sci. Fenn., Ser. A, Vol. 8, 257-324, (1983) · Zbl 0548.30016
[8] Brezis, H.; Fusco, N.; Sbordone, C., Integrability of the Jacobian of orientation preserving mappings, J. Fund. Anal., Vol. 115, 425-431, (1993) · Zbl 0847.26012
[9] Ciarlet, P. G.; Necas, J., Injectivity and self-contact in non-linear elasticity, Arch. Rat. Mech. Anal., Vol. 97, 171-188, (1987) · Zbl 0628.73043
[10] 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, Math Pures Appl., Vol. 72, 1993, pp. 247-286. · Zbl 0684.46044
[11] Dacorogna, B., Direct methods in the calculus of variations, (1989), Springer · Zbl 0703.49001
[12] L. C. Evans and R. E. Gariepy, Lecture Notes on Measure Theory and Fine Properties of Functions, C.R.C. Publ., 1991.
[13] Federer, H., Geometric measure theory, (1969), Springer · Zbl 0176.00801
[14] Flanders, H., Differential forms, (1963), Academic Press · Zbl 0112.32003
[15] Gilbarg, D.; Trudinger, N. S., Elliptic partial differential equations of second order, (1983), Springer · Zbl 0691.35001
[16] Giusti, E., Minimal surfaces and functions of bounded variations, (1984), Birkhäuser
[17] L. Greco and T. Iwaniec, New Inequalities for the Jacobian, Preprint.
[18] T. Iwaniec and A. Lutoborski, Integral Estimates for Null Lagrangians, Preprint, Syracuse University
[19] Iwaniec, T.; Sbordone, C., On the integrability of the Jacobian under minimal hypotheses, Arch. Rat. Mech. Anal., Vol. 119, 129-143, (1992) · Zbl 0766.46016
[20] J. Maly and O. Martio, Lusin’s Condition (N) and Mappings of the Class W^{1, n}, Preprint. · Zbl 0812.30007
[21] J. J. Manfredi, Weakly Monotone Functions, Preprint.
[22] P.A. Meyer, Probability and Potentials, Waltham, 1966 · Zbl 0138.10401
[23] Marcus, M.; Mizel, V. J., Transformations by functions in Sobolev spaces and lower semicontinuity for parametric variational problems, Bull. Amer Math. Soc., Vol. 79, 790-795, (1973) · Zbl 0275.49041
[24] Morrey, C. B., Multiple integrals in the calculus of variations, (1966), Springer · Zbl 0142.38701
[25] McShane, E. J., Integration, (1947), Princeton Univ. press · Zbl 0323.60058
[26] Müller, S.; Surprising, A., Higher integrability property of mappings with positive determinant, Bull. Amer Math. Soc., Vol. 21, 245-248, (1989) · Zbl 0689.49006
[27] Müller, S., Det = det. A remark on the distributional determinant, C.R. Acad. Sci., Paris, Vol. 311, 13-17, (1990) · Zbl 0717.46033
[28] Müller, S., Higher integrability of determinants and weak convergence in L^{1}, J. reine angew. Math., Vol. 412, 20-34, (1990) · Zbl 0713.49004
[29] Müller, S., A counter-example concerning formal integration by parts, C.R. Acad. Sci., Paris, Vol. 312, 45-49, (1991) · Zbl 0723.46028
[30] Müller, S., On the singular support of the distributional determinant, Ann. Inst. H. Poincaré, Analyse non linéaire, Vol. 10, 657-696, (1993) · Zbl 0792.46027
[31] S. Müller and S. J. Spector, Existence Theorems in Nonlinear Elasticity Allowing for Cavitation, submitted to Arch. Rat. Mech. Anal.
[32] S. Müller, S. J. Spector and Q. Tang, Invertibility and a Topological Property of Sobolev Maps (in preparation).
[33] Necas, J., LES méthodes directes en théorie des equations elliptiques, (1967), Masson · Zbl 1225.35003
[34] Ogden, R. W., Large deformation isotropic elasticity - on the correlation of theory and experiment for incompressible rubber-like solids, Proc. Roy. Soc. London, Vol. A326, 565-584, (1972) · Zbl 0257.73034
[35] Reshetnyak, Y. G., On the stability of conformal mappings in multidimensional spaces, Siberian Math. J., Vol. 8, 65-85, (1967)
[36] Reshetnyak, Y. G., Space mappings with bounded distorsion, transl. math. monographs, Ann. Math. Soc., Vol. 73, (1989)
[37] Rickman, S., Quasiregular mappings, (1993), Springer · Zbl 0796.30018
[38] Schwartz, J. T., Nonlinear functional analysis, (1969), Acad. Press · Zbl 0203.14501
[39] Simon, L., Lectures on geometric measure theory, Centre Math Anal., (1983), Australian National University · Zbl 0546.49019
[40] Šverák, V., Regularity properties of deformations with finite energy, Arch. Rat. Mech. Anal., Vol. 100, 105-127, (1988) · Zbl 0659.73038
[41] Tang, Q., Almost-everywhere injectivity in nonlinear elasticity, Proc. Roy. Soc. Edinburgh, Vol. 109A, 79-95, (1988) · Zbl 0656.73010
[42] Vodopyanov, S. K.; Goldstein, V. M., Quasiconformal mappings and spaces of functions with generalised first derivatives, Siberian Math. J., Vol. 12, 515-531, (1977)
[43] Zhang, K. W., Biting theorems for Jacobians and their applications, Ann. Inst. H. Poincaré, Analyse non linéaire, Vol. 7, 345-365, (1990) · Zbl 0717.49012
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