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In this paper, we use $\Gamma$-convergence techniques to study the following variational problem $$S_{\epsilon }^F\left(\Omega \right):=sup\left\{{\epsilon }^{-2^{*}}{\int }_{\Omega }F\left(u\right)dx:{\int }_{\Omega }{\left|\nabla u\right|}^{2}dx\le {\epsilon }^2,u=0\mathrm{on}\partial \Omega \right\},$$ where $0\le F\left(t\right)\le {\left|t\right|}^{2^{*}}$, with $2^{*}=\frac{2n}{n-2}$, and $\Omega$ is a bounded domain of ${ℝ}^n$, $n\ge 3$. We obtain a $\Gamma$-convergence result, on which one can easily read the usual concentration phenomena arising in critical growth problems. We extend the result to a non-homogeneous version of problem $S_{\epsilon }^F\left(\Omega \right)$. Finally, a second order expansion in $\Gamma$-convergence permits to identify the concentration points of the maximizing sequences, also in some...

We study the stability of a sequence of integral functionals on divergence-free matrix valued fields following the direct methods of -convergence. We prove that the -limit is an integral functional on divergence-free matrix valued fields. Moreover, we show that the -limit is also stable under volume constraint and various type of boundary conditions.

We deduce a macroscopic strain gradient theory for plasticity from a model of discrete dislocations. We restrict our analysis to the case of a cylindrical symmetry for the crystal under study, so that the mathematical formulation will involve a two-dimensional variational problem. The dislocations are introduced as point topological defects of the strain fields, for which we compute the elastic energy stored outside the so-called core region. We show that the $\Gamma$-limit of this energy (suitably rescaled),...