zbMATH — the first resource for mathematics

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
Full Text: DOI Numdam EuDML
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.