Rate of Convergence in Periodic Homogenization of Hamilton–Jacobi Equations: The Convex Setting

  • Hiroyoshi Mitake
  • Hung V. TranEmail author
  • Yifeng Yu


We study the rate of convergence of \({u^\varepsilon}\), as \({\varepsilon \to 0+}\), to u in periodic homogenization of Hamilton–Jacobi equations. Here, \({u^\varepsilon}\) and u are viscosity solutions to the oscillatory Hamilton–Jacobi equation and its effective equation
$$\left.\begin{array}{ll}{\rm (C)_\varepsilon}\qquad\begin{cases}u_t^{\varepsilon}+H\left(\frac{x}{\varepsilon}, Du^{\varepsilon}\right) = 0 \qquad & {\rm in} \, \mathbb{R}^{n} \times (0, \infty),\\ u^{\varepsilon}(x, 0) = g(x) \qquad & {\rm on} \, \mathbb{R}^{n},\end{cases}\end{array}\right.$$
$$\left.\begin{array}{ll}{\rm (C)} \qquad \begin{cases}u_t+\overline{H} \left(Du\right)=0 \qquad & {\rm in} \, \mathbb{R}^{n} \times (0, \infty),\\ u(x, 0) = g(x) \qquad & {\rm on} \, \mathbb{R}^{n},\end{cases}\end{array}\right.$$
respectively. We assume that the Hamiltonian HH(y, p) is coercive and convex in the p variable and is \({\mathbb{Z}^n}\)-periodic in the y variable, and the initial data g is bounded and Lipschitz continuous. Here, \({\overline{H}}\) is the effective Hamiltonian.
We prove that
  1. (i)
    $$u^{\varepsilon}(x, t) \geqq u(x, t)- C\varepsilon \quad {{\rm for all} \, (x, t)\in \mathbb{R}^{n} \times [0,\infty)},$$
    where C depends only on H and \({\|Dg\|_{L^\infty(\mathbb{R}^n)}}\) ;
  1. (ii)
    For fixed \({(x, t) \in \mathbb{R}^{n} \times (0, \infty)}\), if u is differentiable at (x, t) and \({\overline{H}}\) is twice differentiable at \({p = Du(x,t)}\), then
    $$u^\varepsilon(x, t) \leqq u(x, t) + \widetilde{C}_{p} t{\varepsilon} + C\varepsilon,$$
    provided that \({g \in C^{2}(\mathbb{R}^n)}\) with \({\|g\|_{C^{2}(\mathbb{R}^n)} < \infty}\). The constant \({\widetilde{C}_p}\) depends only on \({H, \overline{H}, p}\) and g. If g is only Lipschitz continuous, then the above inequality in (ii) is changed into \({u^{\varepsilon}(x, t) \leqq u(x, t) + C_{p} \sqrt{t\varepsilon} + C\varepsilon}\).
When n = 2 and H is positively homogeneous in p of some fixed degree \({k \geqq 1}\), utilizing the Aubry–Mather theory, we obtain the optimal convergence rate \({O(\varepsilon)}\)
$$|u^{\varepsilon}(x, t)-u(x, t) | \leqq C\varepsilon \quad {{\rm for all}\, (x, t)\in \mathbb{R}^2\times [0, \infty).}$$

Here C depends only on H and \({\|Dg\|_{L^{\infty}(\mathbb{R}^2)}}\).

When n = 1, the optimal convergence rate \({O(\varepsilon)}\) is established for any coercive and convex H.

The convergence rate turns out to have deep connections with the dynamics of the underlying Hamiltonian system and the shape of the effective Hamiltonian \({\overline{H}}\). Some related results and counter-examples are obtained as well.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



We are deeply thankful to Hitoshi Ishii, who provides us invaluable comments and suggestions, which help much in vastly improving the presentation of the paper. We also would like to thank Weinan E and Jinxin Xue for helpful comments and discussions.


  1. 1.
    Armstrong, S.N., Cardaliaguet, P., Souganidis, P.E.: Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations. J. Am. Math. Soc. 27(2), 479–540 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations, Second edition, Grundlehren der mathematischen Wissenschaften, Vol. 250, Springer, 1988Google Scholar
  3. 3.
    Bangert, V.: Mather Sets for Twist Maps and Geodesics on Tori, Dynamics Reported, Vol. 1, 1988Google Scholar
  4. 4.
    Bangert, V.: Minimal geodesics. Ergod. Th. Dyn. Syst. 10, 263–286 (1989)MathSciNetGoogle Scholar
  5. 5.
    Bangert, V.: Geodesic rays, Busemann functions and monotone twist maps. Calc. Var. Partial Differ. Equ. 2(1), 49–63 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bernard, P.: The asymptotic behaviour of solutions of the forced Burgers equation on the circle. Nonlinearity 18, 101–124 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Camilli, F., Cesaroni, A., Marchi, C.: Homogenization and vanishing viscosity in fully nonlinear elliptic equations: rate of convergence estimates. Adv. Nonlinear Stud. 11(2), 405–428 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Carneiro, M.J.: On minimizing measures of the action of autonomous Lagrangians. Nonlinearity 8, 1077–1085 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Capuzzo-Dolcetta, I., Ishii, H.: On the rate of convergence in homogenization of Hamilton-Jacobi equations. Indiana Univ. Math. J. 50(3), 1113–1129 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Weinan, E.: Aubry-Mather theory and periodic solutions of the forced Burgers equation. Commun. Pure Appl. Math. 52(7), 811–828 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Evans, L.C.: Periodic homogenisation of certain fully nonlinear partial differential equations. Proc. R. Soc. Edinburgh Sect. A 120(3–4), 245–265 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Evans, L.C., Gomes, D.: Effective Hamiltonians and Averaging for Hamiltonian Dynamics. I. Arch. Ration. Mech. Anal. 157(1), 1–33 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fathi, A.: Weak KAM Theorem in Lagrangian Dynamics Google Scholar
  14. 14.
    Gomes, D.A.: Viscosity solutions of Hamilton-Jacobi equations, and asymptotics for Hamiltonian systems. Calc. Var. 14, 345–357 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hedlund, G.A.: Geodesies on a two-dimensional Riemannian manifold with periodic coefficients. Ann. Math. 33, 719–739 (1932)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lions, P.-L., Papanicolaou, G., Varadhan, S.R.S.: Homogenization of Hamilton–Jacobi Equations, unpublished work, 1987Google Scholar
  17. 17.
    Luo, S., Yu, Y., Zhao, H.: A new approximation for effective Hamiltonians for homogenization of a class of Hamilton-Jacobi equations. Multiscale Model. Simul. 9(2), 711–734 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mitake, H., Tran, H.V.: Homogenization of weakly coupled systems of Hamilton-Jacobi equations with fast switching rates. Arch. Ration. Mech. Anal. 211(3), 733–769 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesUniversity of TokyoMeguro-kuJapan
  2. 2.Department of MathematicsUniversity of Wisconsin MadisonMadisonUSA
  3. 3.Department of MathematicsUniversity of California at IrvineIrvineUSA

Personalised recommendations