Homogenization and two-scale convergence in the Sobolev space with an oscillating exponent

Abstract:

Two-scale convergence in a Sobolev–Orlicz space on a bounded domain $\Omega \subset \mathbb {R}^d$ is explored, a variable exponent $p_\varepsilon (x)=p(x/\varepsilon )$ being a measurable $\varepsilon$-periodic function, $\varepsilon {\in }(0,1]$. Lavrent′ev’s phenomenon occurring in this space is taken into account.

A sequence of potential fields $\nabla u^\varepsilon$ in $L^{p_\varepsilon ( \cdot )}(\Omega )$ is considered under the condition that the norms ${\|\nabla u^\varepsilon \|_{L^{p_\varepsilon ( \cdot )}(\Omega )}}$ are uniformly bounded, and a structure theorem for a two-scale limit of this sequence is proved. A similar structure theorem is proved for a two-scale limit of a sequence of solenoidal fields. These results provide a basis for justification of a homogenization procedure for monotone equations of the form $\mathrm {div} A(x/\varepsilon ,\nabla u^\varepsilon )=\mathrm {div} F$, where the symbol $A(x/\varepsilon ,\xi )$ is $\varepsilon$-periodic over the space variable and, in $\xi$, satisfies coerciveness and boundedness conditions of power type with the exponent $p_\varepsilon (x)$. Such type equations arise in well-known models of electrorheological and thermorheological fluids or the thermistor model.