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Γ -convergence of concentration problems

Micol AmarAdriana Garroni — 2003

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper, we use Γ -convergence techniques to study the following variational problem S ε F ( Ω ) : = sup ε - 2 * Ω F ( u ) d x : Ω | u | 2 d x ε 2 , u = 0 on Ω , where 0 F ( t ) | t | 2 * , with 2 * = 2 n n - 2 , and Ω is a bounded domain of n , n 3 . We obtain a Γ -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 ε F ( Ω ) . Finally, a second order expansion in Γ -convergence permits to identify the concentration points of the maximizing sequences, also in some...

-convergence of functionals on divergence-free fields

Nadia AnsiniAdriana Garroni — 2007

ESAIM: Control, Optimisation and Calculus of Variations

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.

Gradient theory for plasticity via homogenization of discrete dislocations

Adriana GarroniGiovanni LeoniMarcello Ponsiglione — 2010

Journal of the European Mathematical Society

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 Γ -limit of this energy (suitably rescaled),...

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