Calculus of Variations and Geometric Measure Theory

N. Ansini - G. Dal Maso - C. I. Zeppieri

New results on $\Gamma$-limits of integral functionals

created by zeppieri on 04 Mar 2012
modified by ansini on 15 Feb 2014


Published Paper

Inserted: 4 mar 2012
Last Updated: 15 feb 2014

Journal: Ann. Inst. H. Poincaré Anal. Non Linéaire
Volume: 31
Pages: 185--202
Year: 2014


For $\psi\in W^{1,p}(\Omega; \mathbb{R}^{m})$ and $g\in W^{-1,p}(\Omega;\mathbb{R}^d)$, $1< p< +\infty$, we consider a sequence of integral functionals $F^{\psi,g}_k \colon W^{1,p}(\Omega ;\mathbb{R}^m)\times L^{p}(\Omega;\mathbb{R}^{d\times n}) \to [0,+\infty]$ of the form

\[ F^{\psi,g}_k(u,v)=\begin{cases} \int_\Omega f_k(x,\nabla u,v)\,dx & \text{if } u-\psi \in W^{1,p}_0(\Omega;\mathbb{R}^m) \hbox{ and } \; {\rm div} v=g, \cr +\infty & \text{otherwise,} \end{cases} \]

where the integrands $f_k$ satisfy growth conditions of order $p$, uniformly in $k$. We prove a $\Gamma$-compactness result for $F^{\psi,g}_k$ with respect to the weak topology of $W^{1,p}(\Omega;\mathbb{R}^m)\times L^{p}(\Omega;\mathbb{R}^{d\times n})$ and we show that under suitable assumptions the integrand of the $\Gamma$-limit is continuously differentiable. We also provide a result concerning the convergence of momenta for minimizers of $F^{\psi,g}_k$.

Keywords: $\Gamma$-convergence, integral functionals, localization method, $({\rm curl},{\rm div})$-quasiconvexity, convergence of minimizers, convergence of momenta