Dimensional reduction for supremal functionals (Q414765)

The paper is devoted to the study of some dimension reduction problems within the framework of the so-called \(L^\infty\) (or supremal) functionals, i.e. functionals which are represented as \(F(u)=\text{ess}\sup f(x,u(x),Du(x))\), \(x\in \Omega\), where \(\Omega\) is a bounded open set of \({\mathbb R}^N\) and \(u\in W^{1,\infty}(\Omega)\). The introduction of such functionals becomes useful in order to give a mathematical model for many physical problems as, for example, the problem of modeling the dielectric breakdown for a composite conductor.NEWLINENEWLINEThe authors show that the notion of \(\Gamma\)-convergence is not well adapted to supremal functionals. To overcome difficulties connected with it they suggest to use a generalized notion called \(\Gamma^*\)-convergence, and apply the theory on \(\Gamma^*\)-convergence of supremal functionals to a 3D-2D dimensional reduction problem. On the basis of an abstract representation theorem for the \(\Gamma^*\)-limit of a family of supremal functionals it is shown that the limit functional still admits a supremal representation, and a precise identification of its density in some particular cases is given. In particular, the specific form of the \(\Gamma^*\)-limit when dimension reduction is coupled to periodic homogenization, a homogeneous case and a parametrized homogenization are treated.