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

top
In this paper we prove that every weak and strong local minimizer u W 1 , 2 ( Ω , 3 ) of the functional I ( u ) = Ω | D u | 2 + f ( Adj D u ) + g ( det D u ) , where u : Ω 3 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

top

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

top
  1. E. Acerbi and N. Fusco, A regularity theorem for minimizers of quasiconvex integrals. Arch. Rational Mech. Anal.99 (1987) 261–281.  Zbl0627.49007
  2. E. Acerbi and N. Fusco, Regularity for minimizers of non-quadratic functionals: the case 1 < p < 2 . J. Math. Anal. Appl.140 (1989) 115–135.  Zbl0686.49004
  3. J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal.63 (1977) 337–403.  Zbl0368.73040
  4. J.M. Ball, Some open problem in elasticity, in Geometry, Mechanics and dynamics, Springer, New York (2002) 3–59.  Zbl1054.74008
  5. M. Carozza, N. Fusco and G. Mingione, Partial regularity of minimizers of quasiconvex integrals with subquadratic growth. Annali Mat. Pura Appl.175 (1998) 141–164.  Zbl0960.49025
  6. M. Carozza and A. Passarelli di Napoli, A regularity theorem for minimizers of quasiconvex integrals the case 1 < p < 2 . Proc. Roy. Soc. Edinburgh126A (1996) 1181–1199.  Zbl0955.49021
  7. M. Carozza and A. Passarelli di Napoli, Partial regularity of local minimizers of quasiconvex integrals with sub-quadratic growth. Proc. Roy. Soc Edinburgh133A (2003) 1249–1262.  Zbl1059.49033
  8. B. Dacorogna, Direct methods in the calculus of variations. Appl. Math. Sci.78, Springer Verlag (1989).  Zbl0703.49001
  9. L.C. Evans, Quasiconvexity and partial regularity in the calculus of variations. Arch. Rational Mech. Anal.95 (1986) 227–252.  Zbl0627.49006
  10. N. Fusco and J. Hutchinson, Partial regularity in problems motivated by nonlinear elasticity. SIAM J. Math.22 (1991) 1516–1551.  Zbl0744.35014
  11. N. Fusco and J. Hutchinson, Partial regularity and everywhere continuity for a model problem from nonlinear elasticity. J. Australian Math. Soc.57 (1994) 149–157.  Zbl0864.35032
  12. M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems. Ann. Math. Stud.105 Princeton Univ. Press (1983).  Zbl0516.49003
  13. M. Giaquinta and G. Modica, Partial regularity of minimizers of quasiconvex integrals. Ann. Inst. H. Poincaré, Analyse non linéaire3 (1986) 185–208.  Zbl0594.49004
  14. E. Giusti, Metodi diretti in calcolo delle variazioni. U.M.I. (1994).  
  15. J. Kristensen and A. Taheri, Partial regularity of strong local minimizers in the multidimensional calculus of variations. Arch. Rational Mech. Anal.170 (2003) 63–89.  Zbl1030.49040
  16. A. Passarelli di Napoli, A regularity result for a class of polyconvex functionals. Ricerche di MatematicaXLVIII (1999) 379–393.  Zbl0947.35052

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.