Model problems from nonlinear elasticity : partial regularity results

Carozza, Menita; Passarelli Di Napoli, Antonia

ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 1, pp. 120-134.

In this paper we prove that every weak and strong local minimizer $u\in W^{1,2}(\Omega ,{ℝ}^3)$ of the functional $I\left(u\right)={\int }_{\Omega }{\left|Du\right|}^2+f\left(\mathrm{Adj}Du\right)+g\left(\det Du\right),$ where $u:\Omega \subset {ℝ}^3\to {ℝ}^3$, $f$ grows like ${\left|\mathrm{Adj}Du\right|}^p$, $g$ grows like ${\left|\det Du\right|}^q$ and $1<q<p<2$, is $C^{1,\alpha }$ on an open subset ${\Omega }_0$ of $\Omega$ such that $\mathrm{𝑚𝑒𝑎𝑠}(\Omega \setminus {\Omega }_0)=0$. 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.

Classification : 35J50, 35J60, 73C50
Mots clés : nonlinear elasticity, partial regularity, polyconvexity

[1] E. Acerbi and N. Fusco, A regularity theorem for minimizers of quasiconvex integrals. Arch. Rational Mech. Anal. 99 (1987) 261-281. | Zbl

[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. | Zbl

[3] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977) 337-403. | Zbl

[4] J.M. Ball, Some open problem in elasticity, in Geometry, Mechanics and dynamics, Springer, New York (2002) 3-59. | Zbl

[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. | Zbl

[6] M. Carozza and A. Passarelli Di Napoli, A regularity theorem for minimizers of quasiconvex integrals the case $1<p<2$. Proc. Roy. Soc. Edinburgh 126A (1996) 1181-1199. | Zbl

[7] M. Carozza and A. Passarelli Di Napoli, Partial regularity of local minimizers of quasiconvex integrals with sub-quadratic growth. Proc. Roy. Soc Edinburgh 133A (2003) 1249-1262. | Zbl

[8] B. Dacorogna, Direct methods in the calculus of variations. Appl. Math. Sci. 78, Springer Verlag (1989). | MR | Zbl

[9] L.C. Evans, Quasiconvexity and partial regularity in the calculus of variations. Arch. Rational Mech. Anal. 95 (1986) 227-252. | Zbl

[10] N. Fusco and J. Hutchinson, Partial regularity in problems motivated by nonlinear elasticity. SIAM J. Math. 22 (1991) 1516-1551. | Zbl

[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. | Zbl

[12] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems. Ann. Math. Stud. 105 Princeton Univ. Press (1983). | MR | Zbl

[13] M. Giaquinta and G. Modica, Partial regularity of minimizers of quasiconvex integrals. Ann. Inst. H. Poincaré, Analyse non linéaire 3 (1986) 185-208. | Numdam | Zbl

[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. | Zbl

[16] A. Passarelli Di Napoli, A regularity result for a class of polyconvex functionals. Ricerche di Matematica XLVIII (1999) 379-393. | Zbl