Steklov Institute of Mathematics at St.Petersburg

PREPRINT 19/2004

Lower semicontinuity of some functionals under the PDE constraints: $\CalA$-quasiconvex pair

This preprint was accepted November 10, 2004

ABSTRACT:
The problem of establishing necessary and
sufficient conditions for lower semicontinuity under the PDE
constraints is studied for some special class of functionals:
$(u,v,\backslash chi)\backslash mapsto\backslash int\_\backslash Omega\; \backslash bigg\backslash \{\backslash chi(x)\backslash cdot\; F^+(x,u(x),v(x))+(1-\backslash chi(x))\backslash cdot\; F^-(x,u(x),v(x))\backslash bigg\backslash \}dx,$
where $F^\backslash pm:\backslash Omega\backslash times\backslash mathbb\{R\}^m\backslash times\backslash mathbb\{R\}^d\backslash to\backslash mathbb\{R\}$ are  
normal
integrands, $\Omega\subset \mathbb{R}^N$ is a bounded domain, with respect
to the convergence $u_n\to u$ in measure, $v_n\rightharpoonup v$
in $L_p(\Omega;\mathbb{R}^d),\ \mathcal{A}v_n\to 0$ in $W^{-1,p}(\Omega)$
and $\chi_n\rightharpoonup\chi$ in $L_p(\Omega)$, where $\chi_n\in
Z:=\{\chi\in L_\infty(\Omega)\ :\ 0\leq\chi(x)\leq 1,\ a.e.\ x\}$.
Here $\mathcal{A}v=\sum_{i=1}^N A^{(i)}\frac{\partial v}{\partial
x_i}$ is a constant rank partial differential operator.
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