High-frequency homogenization of multidimensional hyperbolic equations

Abstract

In L 2 ( R d ) , we consider an elliptic differential operator A ϵ , ϵ > 0 , of the form A ϵ = − div ⁡ g ( x / ϵ ) ∇ + ϵ − 2 V ( x / ϵ ) with periodic coefficients. For hyperbolic equations with the operator A ϵ , analogs of homogenization problems related to an arbitrary point of the dispersion relation of the operator A 1 are studied (the so-called high-frequency homogenization). For the solutions of the Cauchy problems for these equations with special initial data, approximations in L 2 ( R d ) -norm for small ε are obtained.