Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-05T06:54:43.707Z Has data issue: false hasContentIssue false

Partial regularity and everywhere continuity for a model problem from non-linear elasticity

Published online by Cambridge University Press:  09 April 2009

Nicola Fusco
Affiliation:
Dipartmento di Mathematica “R. Caccioppoli”, Università di Napoli—Monte S. Angelo, Edificio T, via Cintia, 80100 Napoli, Italy
John E. Hutchinson
Affiliation:
Department of Mathematics, School of Mathematical Sciences, Australian National University, GPO Box 4, Canberra ACT 0200, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove a new energy or Caccioppoli type estimate for minimisers of the model functional ∫Ω|Du|2 + (det Du)2, where Ω ⊂ 2 and u: Ω → 2. We apply this to establish C regularity for minimisers except on a closed set of measure zero. We also prove a maximum principle and use this to establish everywhere continuity of minimisers.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Acerbi, E. and Fusco, N., ‘A regularity theorem for minimisers of quasi-convex integrals’, Arch. Rational Mech. Anal. 99 (1981), 261281.CrossRefGoogle Scholar
[2]Ball, J. M., ‘Constitutive inequalities and existence theorems in nonlinear elastostatics’, in: Nonlinear analysis and mechanics: Heriot-Watt symposium, I (ed. Knops, R. J.) (Pitman, London, 1977).Google Scholar
[3]Ball, J. M., ‘Convexity conditions and existence theorems in nonlinear elasticity’, Arch. Rational Mech. Anal. 63 (1977), 337403.CrossRefGoogle Scholar
[4]Evans, L. C., ‘Quasiconvexity and partial regularity in the calculus of variations’, Arch. Rational Mech. Anal. 95 (1986), 227252.CrossRefGoogle Scholar
[5]Evans, L. C. and Gariepy, R. F., ‘Blow-up, compactness and partial regularity in the calculus of variations’, Indiana Univ. Math. J. 36 (1987), 361371.CrossRefGoogle Scholar
[6]Fusco, N. and Hutchinson, J. E., ‘C 1,α partial regularity of functions minimising quasi-convex integrals’, Manuscripta Math. 54 (1985), 121143.CrossRefGoogle Scholar
[7]Fusco, N. and Hutchinson, J. E., ‘Partial regularity in problems motivated by non-linear elasticity’, SIAM J. Math. Anal. 22 (1991), 15161551.CrossRefGoogle Scholar
[8]Giaquinta, M., ‘Multiple integrals in the calculus of variations and nonlinear elliptic systems’, volume 105 Annals of Math. Studies (Princeton Univ. Press, Princeton, 1983).Google Scholar
[9]Giaquinta, M. and Modica, G., ‘Partial regularity of minimizers of quasi-convex integrals’, Ann. Inst. H. Poincaré. Anal. Non Linéaire 3 (1986), 185208.CrossRefGoogle Scholar
[10]Leonetti, F., ‘Maximum principles for functionals depending on minors of the Jacobian matrix of vector-valued mappings’, Australian National University, preprint CMA-R20-1990.Google Scholar
[11]Leonetti, F., ‘Maximum principles for vector-valued minimizers of some integral functionals’, Boll. Un. Mat. Ital. A (6) 5 (1991), 5156.Google Scholar