Deterministic homogenization of integral functionals with convex integrands

Abstract

Abstract. In order to widen the scope of the applications of deterministic homogenization, we consider here the homogenization problem for a family of integral functionals. The homogenization procedure tending to be classical, the choice focused on the convex integral functionals is made just to simplify the presentation of the paper. We use a new approach based on the Stepanov type spaces, which approach allows us to solve various problems such as the almost periodic homogenization problem and others without resorting to additional assumptions. We then apply it to obtain a general homogenization result and then we provide a number of physical applications of the result. The convergence method used falls within the scope of two-scale convergence. Mathematics Subject Classification (2000). 35B40, 45G10, 46J10.

Homogenization in algebras with mean value

“…In that direction we refer, e.g., to the papers [15,30,32,33,34,54,58,59,56,31] in which only ergodic algebras are considered. In this paper we show how one can derive general homogenization results in algebras with mean value through the theory of strongly continuous groups of transformation.…”

“…This issue has already been addressed in many papers (see in particular [3,6,22,31]). In [31] the general deterministic homogenization of (5.1) is addressed, but in separable ergodic algebras wmv using the Σ-convergence method. Here no ergodicity assumption is made on the algebra A and moreover, we use the Young measures theory to solve the problem.…”

“…This reduces considerably the length of this section in contrast to what has been done so far; see e.g. [31]. So we mean here to provide by means of Young measures generated by an algebra wmv, a full study of the functional F ε in the general framework of algebras wmv.…”

Stochastic two-scale convergence of an integral functional

“…It is worth mentioning that the papers [1,17,34,35,39] deal with some interesting problems of stochastic homogenization, but using the monoscale approach. We also mention the paper [25] in which a general deterministic homogenization of convex integral functionals is considered, but still using the monoscale approach.…”

“…, so that, as in the framework of deterministic homogenization (see particularly [25]) one can take this function as a test function in (3.1) (see Definition 1) and get, as E ε → 0, …”