Steklov Institute of Mathematics at St.Petersburg

PREPRINT 19/2004

Alexey Demyanov

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

This preprint was accepted November 10, 2004

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,\chi)\mapsto\int_\Omega \bigg\{\chi(x)\cdot
F^+(x,u(x),v(x))+(1-\chi(x))\cdot F^-(x,u(x),v(x))\bigg\}dx,

where F^\pm:\Omega\times\mathbb{R}^m\times\mathbb{R}^d\to\mathbb{R} are  
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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