Relaxation of quasilinear elliptic systems via A-quasiconvex envelopes

Raitums, Uldis

ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002), pp. 309-334

We consider the weak closure $WZ$ of the set $Z$ of all feasible pairs (solution, flow) of the family of potential elliptic systems

$$\begin{array}{c}\mathrm{div}\left(\sum _{s=1}^{s_0}{\sigma }_s\left(x\right)F_s^{\text{'}}(\nabla u\left(x\right)+g\left(x\right))-f\left(x\right)\right)=0\text{in}\Omega ,\\ u=(u_1,\cdots ,u_m)\in H_0^1(\Omega ;{𝐑}^m),\sigma =({\sigma }_1,\cdots ,{\sigma }_{s_0})\in S,\end{array}$$

where $\Omega \subset {𝐑}^n$ is a bounded Lipschitz domain, $F_s$ are strictly convex smooth functions with quadratic growth and $S=\{\sigma \text{measurable}\mid {\sigma }_s\left(x\right)=0\text{or}1,s=1,\cdots ,s_0,{\sigma }_1\left(x\right)+\cdots +{\sigma }_{s_0}\left(x\right)=1\}$. We show that $WZ$ is the zero level set for an integral functional with the integrand $Qℱ$ being the $𝐀$-quasiconvex envelope for a certain function $ℱ$ and the operator $𝐀={\left(\text{curl,div}\right)}^m$. If the functions $F_s$ are isotropic, then on the characteristic cone $\Lambda$ (defined by the operator $𝐀$) $Qℱ$ coincides with the $𝐀$-polyconvex envelope of $ℱ$ and can be computed by means of rank-one laminates.

Classification : 49J45
Keywords: quasilinear elliptic system, relaxation, A-quasiconvex envelope

[1] J. Ball, B. Kirchheim and J. Kristensen, Regularity of quasiconvex envelopes, Preprint No. 72/1999. Max-Planck Institute für Mathematik in der Naturwissenschaften, Leipzig (1999). | Zbl | MR

[2] B. Dacorogna, Direct Methods in the Calculus of Variations. Springer: Berlin, Heidelberg, New York (1989). | Zbl | MR

[3] I. Fonseca and S. Müller, A-quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal. 30 (1999) 1355-1390. | Zbl | MR

[4] R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems, Parts I-III. Comm. Pure Appl. Math. 39 (1986) 113-137, 138-182, 353-377. | Zbl

[5] K.A. Lurie, A.V. Fedorov and A.V. Cherkaev, Regularization of optimal problems of design of bars and plates, Parts 1 and 2. JOTA 37 (1982) 499-543. | Zbl | MR

[6] M. Miettinen and U. Raitums, On $C^1$-regularity of functions that define G-closure. Z. Anal. Anwendungen 20 (2001) 203-214. | Zbl | MR

[7] F. Murat, Compacité par compensation : condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant. Ann. Scuola Norm. Super. Pisa 8 (1981) 69-102. | Zbl | MR | Numdam

[8] U. Raitums, Properties of optimal control problems for elliptic equations, edited by W. Jäger et al., Partial Differential Equations Theory and Numerical Solutions. Boca Raton: Chapman & Hall/CRC, Res. Notes in Math. 406 (2000) 290-297. | Zbl | MR

[9] L. Tartar, An introduction to the homogenization method in optimal design. CIME Summer Course. Troia (1998). http://www.math.cmu.edu/cna/publications.html | Zbl

[10] V.V. Zhikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer: Berlin, Hedelberg, New York (1994). | Zbl | MR