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MiS Preprint

A-quasiconvexity, lower semicontinuity and Young measures

Stefan Müller and Irene Fonseca


The notion of $A$-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, $u_n \rightharpoonup u$ in$L^p (\Omega; \mathbb{R}^d)$ ( if $p = +\infty $), and $Av_n \to 0$ in $W^{-1,p} (\Omega)$ ($Av_n =0 $ if $p=+\infty$). Here $Av = \sum\limits^{N}_{i=1} A^{(1)} \frac{\partial v}{\partial x_i}$ 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
$f(v)\leq \int_\Omega f(v+w(x))dx$
for all $v\in \mathbb{R^d}$ and all $w\in C^\infty (Q;\mathbb{R}^d)$ such that $Aw =0$, $\int_\Omega w(x)dx=0$, and w is Q-periodic, $Q:=(0,1)^N$. The characterization of Young measures generated by such sequences $\{v_n\}$ is obtained for $1\leq p\leq +\infty $, thus recovering the well known results for the framework $A=$ curl, i.e. when $v_n =\nabla \varphi_n$ for some $\varphi_n \in W^{1,p} (\Omega;R^m)$, $d=N\times m$. In this case $A$-quasiconvexity reduces to Morrey's notion of quasiconvexity.

MSC Codes:
35D99, 35E99, 49J45
a-quasiconvexity, equi-integrability, young measure, lower semicontinuity

Related publications

1999 Repository Open Access
Irene Fonseca and Stefan Müller

A-quasiconvexity, lower semicontinuity, and Young measures

In: SIAM journal on mathematical analysis, 30 (1999) 6, 1355-1390 (electronic)