MiS Preprint Repository

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 $$v\in L^1(\mathrm{\Omega };{\mathbb{R}}^m)\mapsto {\int }_{\mathrm{\Omega }}f(x,v(x))dx,$$ where admissible sequences $\{v_n\}$ satisfy a first order system of PDEs $\mathcal{A}{v}_n=0$. We suppose that $\mathcal{A}$ has constant rank, $f$\ is $\mathcal{A}$-quasiconvex and satisfies the non standard growth conditions $$\frac{1}{C}(|v{|}^p-1)\le f(v)\le C(1+|v{|}^q)$$ with $q\in [p,pN/(N-1))$ for $p\le N-1$, $q\in [p,p+1)$ for $p>N-1.$ In particular, our results generalize earlier work where $\mathcal{A}v=0$ reduced to $v={\mathrm{\nabla }}^{s}u$ for some $s\in \mathbb{N}$.