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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alvarez O., Bardi M.: Viscosity solutions methods for singular perturbations in deterministic and stochastic control. SIAM J. Control Optim. 40(4), 1159–1188 (2002)
Arisawa M., Lions P.L.: On ergodic stochastic control. Commun. Partial Differ. Equ. 23(11–12), 333–358 (1998)
Bakhvalov, N., Panasenko, G.: Homogenisation averaging processes in periodic media: mathematical problems in the mechanics of composite materials. Mathematics and its Applications Soviet Series. Springer, Berlin 1989
Bensoussan, A., Papanicolau, G., Lions, J.L.: Asymptotic analysis for periodic structures. Elsevier, Amsterdam 1978
Caffarelli L.: A note on nonlinear homogenization. Commun. Pure Appl. Math. 52(7), 829–838 (1999)
Caffarelli, L.A., Cabré, X.: Fully nonlinear elliptic equations, vol. 43. American Mathematical Soc., Providence 1995
Caffarelli L.A., Souganidis P.E.: Rates of convergence for the homogenization of fully nonlinear uniformly elliptic PDE in random media. Invent. Math. 180(2), 301–360 (2010)
Caffarelli L.A., Souganidis P.E., Wang L.: Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media. Commun. Pure Appl. Math. 58(3), 319–361 (2005)
Camilli F., Marchi C.: Rates of convergence in periodic homogenization of fully nonlinear uniformly elliptic PDEs. Nonlinearity 22(6), 1481–1498 (2009)
Crandall M.G., Ishii H., Lions P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27(1), 1–67 (1992)
Dolcetta I.C., Ishii H.: On the rate of convergence in homogenization of Hamilton–Jacobi equations. Indiana Univ. Math. J. 50(3), 1113–1129 (2001)
Evans L.C.: The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. R. Soc. Edinb. Sect. A Math. 111(3–4), 359–375 (1989)
Evans L.C.: Periodic homogenisation of certain fully nonlinear partial differential equations. Proc. R. Soc. Edinb. Sect. A Math. 120(3–4), 245–265 (1992)
Evans L.C., Gomes D.: Effective hamiltonians and averaging for hamiltonian dynamics i. Arch. Ration. Mech. Anal. 157(1), 1–33 (2001)
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, vol. 224. Springer, Berlin 2001
Ishii H.: Almost periodic homogenization of Hamilton–Jacobi equations. Int. Conf. Differ. Equ. 1(2), 600–605 (2000)
Jikov, V.V., Oleinik, O., Kozlov, S.M.: Homogenization of differential operators and integral functionals. Springer, Berlin 1994
Lions, P.L., Papanicolaou, G., Varadhan, S.R.: Homogenization of Hamilton–Jacobi equations. Preliminary version (1988)
Lions P.L., Souganidis P.E.: Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications. Annales de l’Institut Henri Poincare (C) Non Linear Analysis 22(5), 667–677 (2005)
Majda A.J., Souganidis P.E.: Large scale front dynamics for turbulent reaction–diffusion equations with separated velocity scales. Nonlinearity 7(1), 1–30 (1994)
Marchi C.: Continuous dependence estimates for the ergodic problem of bellman equation with an application to the rate of convergence for the homogenization problem. Calc. Var. Partial Differ. Equ. 51(3–4), 539–553 (2014)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. Del Maso
Rights and permissions
About this article
Cite this article
Kim, S., Lee, KA. Higher Order Convergence Rates in Theory of Homogenization: Equations of Non-divergence Form. Arch Rational Mech Anal 219, 1273–1304 (2016). https://doi.org/10.1007/s00205-015-0921-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-015-0921-7