$\cal A$-Quasiconvexity, Lower Semicontinuity, and Young Measures 论文

摘要

The notion of ${\cal A}$-quasiconvexity is introduced as a necessary and sufficient condition for (sequential) lower semicontinuity of $$ (u,v) \mapsto \int_{\Omega} f(x,u(x), v(x))\, dx $$ whenever $f: \Omega\times\mathbb R^m\times \mathbb R^d \to [0,+\infty)$ is a normal integrand, $\Omega\subset \mathbb R^N$ is open, bounded, $u_n \to u$ in measure, $v_n \weak v$ in $L^p(\Omega;\mathbb R^d)$ ($\weakst$ if $p = +\infty$), and ${\cal A} v_n \to 0$ in $W^{-1,p}(\Omega)$ (${\cal A} v_n = 0$ if $p=+\infty$). Here ${\cal A}v = \sum_{i=1}^N A^{(i)} {{\partial v} \over {\partial x_i}}$ is aconstant rank partial differential operator, $A^{(i)} \in {\L}(\mathbb R^d; \mathbb R^l)$, and $f(x,u,\cdot)$ is ${\cal A}$-quasi-convex if $$ f(x,u,v) \leq \int_{Q} f(x,u,v + w(y)) \, dy $$ for all $v \in \mathbb R^d$ and all $w \in C^{\infty}(Q;\mathbb R^d)$ such that ${\cal A} w=0$, $\int_{Q} w(x) \, dx = 0$, and w is Q-periodic, $Q := (0,1)^N$. The characterization of Young measures generated by such sequences $\{v_n\}$ is obtained for $1 \leq p < + \infty $, thus recovering the well-known results for the framework ${\cal A}=$ curl, i.e., when $v_n = \nabla \varphi_n$ for some $\varphi_n \in W^{1,p}(\Omega;\mathbb R^m)$, $d = N \times m$. In this case ${\cal A}$-quasiconvexity reduces to Morrey's notion of quasiconvexity.