MiS Preprint Repository

The notion of $A$-quasiconvexity is introduced as a necessary and sufficient condition for (sequential) lower semicontinuity of
$(u,v)\mapsto {\int }_{\mathrm{\Omega }}f(x,u(x),V(x))dx$
whenever $f:\mathrm{\Omega }\times {\mathbb{R}}^m\times {\mathbb{R}}^d\to [0,+\mathrm{\infty }]$ is a normal integrand, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is open, bounded, $u_n\to u$ in measure, $u_n\rightharpoonup u$ in$L^p(\mathrm{\Omega };{\mathbb{R}}^d)$ ( if $p=+\mathrm{\infty }$), and $A{v}_n\to 0$ in $W^{-1,p}(\mathrm{\Omega })$ ($A{v}_n=0$ if $p=+\mathrm{\infty }$). Here $Av=\sum _{i=1}^N{A}^{(1)}\frac{\partial v}{\partial x_i}$ is a constant-rank partial differential operator, $A(i)\in L({\mathbb{R}}^d;\mathbb{R})$, and $f(x,u,\cdot )$ is $A$-quasiconvex if
$f(v)\le {\int }_{\mathrm{\Omega }}f(v+w(x))dx$
for all $v\in {\mathbb{R}}^{\mathbb{d}}$ and all $w\in C^{\mathrm{\infty }}(Q;{\mathbb{R}}^d)$ such that $Aw=0$, ${\int }_{\mathrm{\Omega }}w(x)dx=0$, and w is Q-periodic, $Q:=(0,1{)}^N$. The characterization of Young measures generated by such sequences $\{v_n\}$ is obtained for $1\le p\le +\mathrm{\infty }$, thus recovering the well known results for the framework $A=$ curl, i.e. when $v_n=\mathrm{\nabla }{\phi }_n$ for some ${\phi }_n\in W^{1,p}(\mathrm{\Omega };R^m)$, $d=N\times m$. In this case $A$-quasiconvexity reduces to Morrey's notion of quasiconvexity.