A-quasiconvexity with variable coefficients

Pedro M. Santos
Department of Mathematical Sciences
Carnegie Mellon University

March 17, 2003

It is shown that for integrals of the type

$\displaystyle I(u,v):=\int_{\Omega} f(x,u(x),v(x))\,dx,$    

with $ \Omega \subset {\mathbb{R}}^N$ open, bounded, and $ f:\Omega \times {\mathbb{R}}^m \times {\mathbb{R}}^d \to [0,+\infty)$ Carathéodory satisfying a growth condition $ 0 \le f(x,u,v) \le C(1+{\vert v\vert}^p)$, for some $ p \in (1,+\infty)$, a sufficient condition for lower semicontinuity along sequences $ u_n \to u$ in measure, $ v_n \rightharpoonup v$ in $ L^p$, $ {\cal A}v_n \to 0$ in $ W^{-1,p}$ is the $ {\cal A}_x$-quasiconvexity of $ f(x,u,.)$. Here $ {\cal A}$ is a variable coefficients operator of the form

$\displaystyle {\cal A}:= \sum_{i=1}^N A^{(i)}(x)\frac {\partial} {\partial x_i},$    

with $ A^{(i)} \in C^{\infty}(\Omega;{\cal M}^{l \times d}) \cap W^{1,\infty}$, $ i=1,..N$, satisfying the condition

$\displaystyle {\rm rank}\left( \sum_{i=1}^N A^{(i)}(x)\omega_i \right)={\rm const}$   for $ x \in \Omega$ and $ \omega \in {\mathbb{R}}^N \setminus \{0\}$$\displaystyle ,$    

and $ {\cal A}_x$ denotes the constant coefficients operator one obtains by freezing $ x$. Under additional regularity conditions on $ f$ it is proved that the condition above is also necessary. A characterization of the Young measures generated by bounded sequences $ \{v_n\}$ in $ L^p$ satisfying the condition $ {\cal A}v_n \to 0$ in $ W^{-1,p}$ is obtained.

Get the paper in its entirety as