# PREPRINT 19/2004

Alexey Demyanov

## 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:
$\left(u,v,\chi\right)\mapsto\int_\Omega \bigg\\left\{\chi\left(x\right)\cdot F^+\left(x,u\left(x\right),v\left(x\right)\right)+\left(1-\chi\left(x\right)\right)\cdot F^-\left(x,u\left(x\right),v\left(x\right)\right)\bigg\\right\}dx,$
where $F^\pm:\Omega\times\mathbb\left\{R\right\}^m\times\mathbb\left\{R\right\}^d\to\mathbb\left\{R\right\}$ 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.
[Full text:
(.ps.gz)]

Back to all preprints

Back to the Steklov
Institute of Mathematics at St.Petersburg