A problem of homogenization of a divergence-type second order uniformly elliptic operator is considered with arbitrary bounded rapidly oscillating periodic coefficients, either with periodic "outer" boundary conditions or in the whole space. It is proved that if the right-hand side is Gevrey regular (in particular, analytic), then by optimally truncating the full two-scale asymptotic expansion for the solution one obtains an approximation with an exponentially small error. The optimality of the exponential error bound is established for a one-dimensional example by proving the analogous lower bound.
Introduction.Classical homogenization theory describes the relation of solutions u ε (x) of boundary value problems with rapidly oscillating coefficients to solutions u 0 (x) of a homogenized problem, i.e., a problem without rapidly oscillating coefficients. In appropriate function spaces convergence can be established as ε → 0 (with ε describing the period or wavelength of the coefficients' oscillations); see, e.g., [1,2,3,4] and the references therein. For particular homogenization problems, e.g., for those described by linear second order elliptic PDEs with periodic coefficients, the rate of convergence with respect to ε can often also be determined, see, e.g., [4,5,6,7]. The order of convergence can sometimes be improved further by constructing higher order correctors. The presence of a boundary creates additional "boundary layers," which substantially complexifies the problem of constructing the higher order terms; see, e.g., [5,3,4,8,9]. However, in the absence of the boundary, either for a problem with outer periodicity conditions or in the whole space (away from the spectrum), higher order terms can often be explicitly constructed. In particular, under the assumptions of sufficient regularity of the coefficients and the right-hand side of the equation, it is possible to construct and rigorously justify a full two-scale asymptotic expansion for u ε (x), i.e., to establish the error bounds both for linear problems (e.g., [3,10]) and even for appropriate nonlinear ones [11].The above can be referred to, in the context of homogenization, as "homogenization in all orders," by analogy with "asymptotics in all orders": by appropriately