Asymptotic expansions for eigenvalues and eigenfunctions of elliptic boundary-value problems with rapidly oscillating coefficients (Q2735868)

It is investigated a question of \(S\)-homogenization of a family of optimal control problems of such types NEWLINE\[NEWLINE\inf I_\varepsilon(x,y), \qquad A_\varepsilon(x,y)= f_\varepsilon, \quad F_\varepsilon(x,y)\geq 0,NEWLINE\]NEWLINE where \(A_\varepsilon\), \(F_\varepsilon\) are nonlinear operators, which may depend on \(\varepsilon\) arbitrary, \(I_\varepsilon\) is a cost function, and \(\varepsilon\) denotes a ``small'' multiparameter with a set \(E\), partially ordered by decreasing \((0\leq \varepsilon\) for every \(\varepsilon\in E\) and 0 is the minimal element in \(E)\). NEWLINENEWLINENEWLINEThe existence of a strongly \(S\)-homogenized optimal control problem is devoted and several important variational and topological properties of it are obtained. The representation formula of the homogenized problem is given.