\(\Gamma\)-convergence through Young measures. (English) Zbl 1077.49012

Pedregal, Pablo

SIAM J. Math. Anal. 36, No. 2, 423-440 (2004).

The paper is concerned with an approach to \(\Gamma\)-convergence through the study of the underlying Young measures associated to relevant sequences, and by using as major tool the slicing measure decomposition.
A first case considered in the paper deals with the asymptotic behaviour of a sequence of functionals of the type \[ I_j(u)=\int_\Omega W(a_j(x),u(x))dx,\quad u\in{\mathcal A}, \] where \(\Omega\) is a bounded regular domain in \({\mathbb R}^N\), \(a_j\colon\Omega\to{\mathbb R}^m\), \(W\colon{\mathbb R}^m\times{\mathbb R}^d\to{\mathbb R}\) is locally uniformly continuous and coercive, and \({\mathcal A}\) is some weakly closed subset of certain reflexive Lebesgue space. By assuming that \(\{a_j\}\) weakly converges in \(L^q(\Omega)\) for some \(q>1\), it is proved that the \(\Gamma\)-limit \(I\) of \(\{I_j\}\) is given by \[ I(u)=\int_\Omega\psi(x,u(x))dx, \] for some explicitly constructed \(\psi\).
The situation where the functionals depend on gradients, namely where \[ I_j(u)=\int_\Omega W(a_j(x),\nabla u(x))dx,\quad u\in{\mathcal A}, \] has also been treated. This time \({\mathcal A}\) is some weakly closed subset of certain reflexive Sobolev space. Under an additional structure assumption on \(\{a_j\}\), it is proved that \[ I(u)=\int_\Omega\psi(x,\nabla u(x))dx, \] for another explicitly constructed \(\psi\).
Some examples are also discussed.

Reviewer: Riccardo De Arcangelis (Napoli)

MSC:

49J45	Methods involving semicontinuity and convergence; relaxation
49J20	Existence theories for optimal control problems involving partial differential equations
74Q05	Homogenization in equilibrium problems of solid mechanics

Keywords:

\(\Gamma\)-convergence; Young measures; slicing decomposition