Calculus of Variations and Geometric Measure Theory

I. Fonseca - G. Leoni - S. Mueller

$A$-quasiconvexity: weak convergence and the gap

created on 25 Sep 2002
modified by leoni on 10 Jan 2005


Published Paper

Inserted: 25 sep 2002
Last Updated: 10 jan 2005

Journal: Ann. Inst. H. Poincaré Anal. Non Linéaire
Volume: 21
Number: 2
Pages: 209-236
Year: 2004


Lower semicontinuity results with respect to weak-$\ast$ convergence in the sense of measures and with respect to weak convergence in $L^p$ are obtained for functionals $$ F(v)=int{Omega}f(x,v(x))\,dx, $$ where admissible sequences $\{v_{n}\}$ satisfy a first order system of PDEs $A v_{n}=0$. We suppose that $A$ has constant rank, $f$ is $A$-quasiconvex and satisfies the non standard growth conditions $$ 1C(
{p}-1)<= f(v)<= C(1+
{q}) $$ with $q$ in $( p,pN/(N-1))$ for $p<=N-1$, while $q$ in $( p,p+1)$ for $p>N-1.$ In particular, our results generalize earlier work where $A v=0$ reduced to $v=D^{s}u$ for some integer $s$.