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Relaxation of Quasilinear Elliptic Systems via A-quasiconvex Envelopes

Published online by Cambridge University Press:  15 September 2002

Uldis Raitums
Uldis Raitums*
Affiliation:
Institute of Mathematics and Computer Science, University of Latvia, 1459 Riga, Latvia; [email protected].
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Abstract

We consider the weak closure WZ of the set Z of all feasible pairs (solution, flow) of the family of potential elliptic systems $$ \begin{array}{c}\mbox{div}\left(\sum\limits_{s=1}^{s_0}\sigma_s(x)F_s^\prime (\nabla u(x)+g(x))-f(x)\right)=0\;\mbox{in}\,\Omega, u=(u_1,\dots, u_m)\in H_0^1(\Omega;{\bf R}^m),\;\sigma=(\sigma_1,\dots,\sigma_{s_0})\in S,\end{array} $$ where Ω ⊂ Rn is a bounded Lipschitz domain, Fs are strictly convex smooth functions with quadratic growth and $S=\{\sigma\, measurable\,\mid\,\sigma_s(x)=0\;\mbox{or}\,1,\;s=1,\dots,s_0,\;\sigma_1(x)+\cdots +\sigma_{s_0}(x)=1\}$. We show that WZ is the zero level set for an integral functional with the integrand $Q\cal F$ being the A-quasiconvex envelope for a certain function $\cal F$ and the operator A = (curl,div)m. If the functions Fs are isotropic, then on the characteristic cone Λ (defined by the operator A) $Q{\cal F}$ coincides with the A-polyconvex envelope of $\cal F$ and can be computed by means of rank-one laminates.

Keywords

Quasilinear elliptic systemrelaxationA-quasiconvex envelope.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 7 , 2002 , pp. 309 - 334
Copyright
© EDP Sciences, SMAI, 2002

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