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Model problems from nonlinear elasticity: partial regularity results

Published online by Cambridge University Press:  14 February 2007

Menita Carozza  and
Antonia Passarelli di Napoli
Menita Carozza
Affiliation:
Dipartimento Pe.Me.Is., Piazza Arechi II, 82100 Benevento, Italy; carozza@unisannio.it
Antonia Passarelli di Napoli
Affiliation:
Dipartimento di Matematica e Appl. “R.Caccioppoli” Universitá di Napoli “Federico II” Via Cintia, 80126 Napoli, Italy; antonia.passarelli@unina.it
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Abstract

In this paper we prove that every weak and strong local minimizer $u\in{W^{1,2}(\Omega,\mathbb{R}^3)}$ of the functional $I(u)=\int_\Omega|Du|^2+f({\rm Adj}Du)+g({\rm det}Du),$ where $ u:\Omega\subset\mathbb{R}^3\to \mathbb{R}^3$, f grows like $|{\rm Adj}Du|^p$, g grows like $|{\rm det}Du|^q$ and 1<q<p<2, is $C^{1,\alpha}$ on an open subset $\Omega_0$ of Ω such that ${\it meas}(\Omega\setminus \Omega_0)=0$. Such functionals naturally arise from nonlinear elasticity problems. The key point in order to obtain the partial regularity result is to establish an energy estimate of Caccioppoli type, which is based on an appropriate choice of the test functions. The limit case $p=q\le 2$ is also treated for weak local minimizers.

Keywords

Nonlinear elasticitypartial regularitypolyconvexity.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 1 , January 2007 , pp. 120 - 134
Copyright
© EDP Sciences, SMAI, 2007

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