Model problems from nonlinear elasticity: partial regularity results

Menita Carozza; Antonia Passarelli di Napoli

ESAIM: Control, Optimisation and Calculus of Variations (2007)

  • Volume: 13, Issue: 1, page 120-134
  • ISSN: 1292-8119

Abstract

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In this paper we prove that every weak and strong local minimizer u ∈ W 1 , 2 ( Ω , ℝ 3 ) of the functional I ( u ) = ∫ Ω | D u | 2 + f ( Adj D u ) + g ( det D u ) , where u : Ω ⊂ ℝ 3 → ℝ 3 , f grows like | Adj D u | p , g grows like | det D u | q and 1<q<p<2, is C 1 , α on an open subset Ω 0 of Ω such that 𝑚𝑒𝑎𝑠 ( Ω ∖ Ω 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 ≤ 2 is also treated for weak local minimizers.

How to cite

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Carozza, Menita, and Passarelli di Napoli, Antonia. "Model problems from nonlinear elasticity: partial regularity results." ESAIM: Control, Optimisation and Calculus of Variations 13.1 (2007): 120-134. <http://eudml.org/doc/250002>.

@article{Carozza2007,
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. },
author = {Carozza, Menita, Passarelli di Napoli, Antonia},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Nonlinear elasticity; partial regularity; polyconvexity.; Nonlinear elasticity; polyconvexity},
language = {eng},
month = {2},
number = {1},
pages = {120-134},
publisher = {EDP Sciences},
title = {Model problems from nonlinear elasticity: partial regularity results},
url = {http://eudml.org/doc/250002},
volume = {13},
year = {2007},
}

TY - JOUR
AU - Carozza, Menita
AU - Passarelli di Napoli, Antonia
TI - Model problems from nonlinear elasticity: partial regularity results
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2007/2//
PB - EDP Sciences
VL - 13
IS - 1
SP - 120
EP - 134
AB - 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.
LA - eng
KW - Nonlinear elasticity; partial regularity; polyconvexity.; Nonlinear elasticity; polyconvexity
UR - http://eudml.org/doc/250002
ER -

References

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