Higher Order Convergence Rates in Theory of Homogenization: Equations of Non-divergence Form

Abstract

We establish higher order convergence rates in the theory of periodic homogenization of both linear and fully nonlinear uniformly elliptic equations of non-divergence form. The rates are achieved by involving higher order correctors which fix the errors occurring both in the interior and on the boundary layer of our physical domain. The proof is based on a viscosity method and a new regularity theory which captures the stability of the correctors with respect to the shape of our limit profile.

Higher order convergence rates in theory of homogenization III: Viscous Hamilton–Jacobi equations

“…In this section, we shall investigate the regularity ofH and w in the slow variable p. Such a regularity has been established in the authors' previous works [KL1] and [KL2], for fully nonlinear elliptic and, respectively, parabolic PDEs. Let us first observe the continuity of w in p variable.…”

“…This paper is in the sequel of the authors' previous works [KL1] and [KL2], where the higher order convergence rates were achieved in the periodic homogenization of fully nonlinear, uniformly elliptic and parabolic, second order PDEs. We found it interesting in the previous works that even if we begin with a nonlinear PDE at the first order approximation, we no longer encounter such a nonlinear structure in the second and the higher order approximations.…”

“…For the rest of this section, we will justify the existence of the higher order interior correctors in a rigorous way. The corresponding work has been done by the authors in [KL1] and [KL2] in the framework of fully nonlinear, uniformly elliptic, second order PDEs in non-divergence form.…”

“…We shall call w k , chosen from Lemma 5.1, the k-th order interior corrector for the homogenization problem (1.1), due to the following lemma. Although the computation involved in the proof below is similar to what can be found in [KL1,Section 3.3] and [KL2, Section 4.1], we present it in detail for the sake of completeness.…”

“…Nevertheless, this is the first work on the higher order convergence rates in the regime of (viscous) Hamilton-Jacobi equations. For the higher order convergence rates for other type of equations, we refer to [KL1], [KL2] and the references therein.…”