Relaxation of quasilinear elliptic systems via A-quasiconvex envelopes
ESAIM: Control, Optimisation and Calculus of Variations, Volume 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

 $\begin{array}{c}\mathrm{div}\left(\sum _{s=1}^{{s}_{0}}{\sigma }_{s}\left(x\right){F}_{s}^{\text{'}}\left(\nabla u\left(x\right)+g\left(x\right)\right)-f\left(x\right)\right)=0\phantom{\rule{0.277778em}{0ex}}\text{in}\phantom{\rule{0.166667em}{0ex}}\Omega ,\\ u=\left({u}_{1},\cdots ,{u}_{m}\right)\in {H}_{0}^{1}\left(\Omega ;{𝐑}^{m}\right),\phantom{\rule{0.277778em}{0ex}}\sigma =\left({\sigma }_{1},\cdots ,{\sigma }_{{s}_{0}}\right)\in S,\end{array}$
where $\Omega \subset {𝐑}^{n}$ is a bounded Lipschitz domain, ${F}_{s}$ are strictly convex smooth functions with quadratic growth and $S=\left\{\sigma \phantom{\rule{4pt}{0ex}}\text{measurable}\phantom{\rule{0.166667em}{0ex}}\mid \phantom{\rule{0.166667em}{0ex}}{\sigma }_{s}\left(x\right)=0\phantom{\rule{0.277778em}{0ex}}\text{or}\phantom{\rule{0.166667em}{0ex}}1,\phantom{\rule{0.277778em}{0ex}}s=1,\cdots ,{s}_{0},\phantom{\rule{0.277778em}{0ex}}{\sigma }_{1}\left(x\right)+\cdots +{\sigma }_{{s}_{0}}\left(x\right)=1\right\}$. 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.

DOI: 10.1051/cocv:2002014
Classification: 49J45
Keywords: quasilinear elliptic system, relaxation, A-quasiconvex envelope
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author = {Raitums, Uldis},
title = {Relaxation of quasilinear elliptic systems via {A-quasiconvex} envelopes},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {309--334},
publisher = {EDP-Sciences},
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zbl = {1037.49011},
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Raitums, Uldis. Relaxation of quasilinear elliptic systems via A-quasiconvex envelopes. ESAIM: Control, Optimisation and Calculus of Variations, Volume 7 (2002), pp. 309-334. doi : 10.1051/cocv:2002014. http://www.numdam.org/articles/10.1051/cocv:2002014/

[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). | MR | Zbl

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

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

[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. | MR | Zbl

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

[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. | Numdam | MR | Zbl

[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. | MR | Zbl

[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). | MR | Zbl

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