Lower semicontinuity of some functionals under the PDE constraints: $\mathcal{A}$-quasiconvex pair

A. V. Demyanov

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 36, Tome 318 (2004), pp. 100-119

Résumé

The problem of establishing necessary and sufficient conditions for l.s.c. under the PDE constraints is studied for some special class of functionals: $$ (u,v,\chi)\mapsto\int_\Omega \biggl\{\chi(x)\cdot F^+(x,u(x),v(x))+(1-\chi(x))\cdot F^-(x,u(x),v(x))\biggr\}\,dx, $$ with respect to the convergence $u_n\to u$ in measure, $v_n\rightharpoonup v$ in $L_p(\Omega;\mathbb{R}^d)$, $\mathcal{A}v_n\to0$ in $W^{-1,p}(\Omega)$ and $\chi_n\rightharpoonup\chi$ in $L_p(\Omega)$, where $\chi_n\in Z:=\{\chi\in L_\infty(\Omega):0\leq\chi(x)\leq1,\text{ a.e. }x\}$. Here $\mathcal{A}v=\sum_{i=1}^N A^{(i)}\frac{\partial v}{\partial x_i}$ is a constant rank partial differential operator. The main result is that if the characteristic cone of $\mathcal{A}$ has the full dimension, then l.s.c. is equivalent to the fact that $F^\pm$ are both $\mathcal{A}$-quasiconvex and for a.e. $x\in\Omega$, for all $u\in\mathbb{R}^d$ $$ F^+(x,u,\cdot\,)-F^-(x,u,\cdot\,)\equiv C(x,u). $$ As a corollary, we obtain the results for the functional $$ (u,v,\chi)\mapsto\int_\Omega\chi(x)\cdot f(x,u(x),v(x))\,dx, $$ with respect to the same convergence. We show, that this functional is l.s.c. iff $$ f(x,u,v)\equiv g(x,u). $$

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