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Global invertibility of Sobolev functions and the interpenetration of matter

Published online by Cambridge University Press:  14 November 2011

J. M. Ball
J. M. Ball
Affiliation:
Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS
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Synopsis

A global inverse function theorem is established for mappings u: Ω → ℝn, Ω ⊂ ℝn bounded and open, belonging to the Sobolev space W1.p(Ω), p > n. The theorem is applied to the pure displacement boundary value problem of nonlinear elastostatics, the conclusion being that there is no interpenetration of matter for the energy-minimizing displacement field.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 88 , Issue 3-4 , 1981 , pp. 315 - 328
Copyright
Copyright © Royal Society of Edinburgh 1981

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References

1Adams, R. A.. Sobolev spaces (New York: Academic, 1975).Google Scholar
2Antman, S. S.. Ordinary differential equations of nonlinear elasticity II: Existence and regularity theory for conservative boundary value problems. Arch. Rational Mech. Anal. 61 (1976), 353–393.CrossRefGoogle Scholar
3Antman, S. S. and Brezis, H.. The existence of orientation-preserving deformations in nonlinear elasticity. In Nonlinear analysis and mechanics: Heriot-Watt Symposium vol. II, ed. Knops, R. J..(London: Pitman, 1978).Google Scholar
4Ball, J. M.. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977), 337–403.CrossRefGoogle Scholar
5Ball, J. M.. Constitutive inequalities and existence theorems in nonlinear elastostatics. In Nonlinear analysis and mechanics: Heriot-Watt Symposium vol. I, ed. Knops, R. J..(London: Pitman, 1977).Google Scholar
6Ball, J. M.. Remarques sur l'existence et la régularité des solutions d'élastostatique nonlinéaire (London: Pitman, to appear).Google Scholar
7Ball, J. M.. Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, to appear.Google Scholar
8Ball, J. M., J. Currie, C. and Olver, P. J.. Null Lagrangians, weak continuity, and variational problems of arbitrary order. J. Functional Analysis, to appear.Google Scholar
9Berger, M. S.. Nonlinearity and Functional Analysis (New York: Academic, 1977).Google Scholar
10Bony, J. M.. Principe du maximum dans les espaces de Sobolev. C.R. Acad. Sci. Paris Sér. A 265 (1967), 333–336.Google Scholar
11Browder, F. E.. Nonlinear Operators and Nonlinear Equations of Evolution in Banach spaces. Proc. Symposia in Pure Mathematics 18, Pt 2 (Providence, R.I.: Amer. Math. Soc, 1976).Google Scholar
12Clarke, F. H.. On the inverse function theorem. Pacific J. Math. 64 (1976), 97–102.CrossRefGoogle Scholar
13Golubitsky, M. and Guillemin, V.. Stable mappings and their singularities (Berlin: Springer, 1973).CrossRefGoogle Scholar
14Marcus, M. and Mizel, V. J.. Transformations by functions in Sobolev spaces and lower semicontinuity for parametric variational problems. Bull. Amer. Math. Soc. 79 (1973), 790–795.CrossRefGoogle Scholar
15Meisters, G. H. and Olech, C.. Locally one-to-one mappings and a classical theorem on Schlicht functions. Duke Math. J. 30 (1963), 63–80.CrossRefGoogle Scholar
16Morrey, C. B.. Multiple integrals in the calculus of variations (Berlin: Springer, 1966).CrossRefGoogle Scholar
17Nirenberg, L.. Topics in nonlinear functional analysis (New York: Courant Institute Lecture Notes, 1974).Google Scholar
18Rado, T. and Reichelderfer, P. V.. Continuous transformations in analysis (Berlin: Springer, 1955).CrossRefGoogle Scholar
19Schwartz, J. T.. Nonlinear functional analysis (New York: Gordon and Breach, 1969).Google Scholar
20Vodop'yanov, S. K. and Gol'dshtein, V. M.. Quasiconformal mappings and spaces of functions with generalized first derivatives. Sibirsk. Mat. Z. 17 (1976), 515–531.Google Scholar
21Vodop'yanov, S. K., Gol'dshtein, V. M. and Reshetnyak, Yu. G.. On geometric properties of functions with generalized first derivatives. Russian Math. Surveys 34 (1979), 19–74.CrossRefGoogle Scholar