We study the optimal convergence rate for the homogenization problem of convex Hamilton–Jacobi equations when the Hamitonian is periodic with respect to spatial and time variables, and notably time-dependent. We prove a result similar to that of (Tran and Yu (2021)), which means the optimal convergence rate is also .
I.Capuzzo-Dolcetta and H.Ishii, On the rate of convergence in homogenization of Hamilton–Jacobi equations, Indiana Univ. Math. J.50(3) (2001), 1113–1129. doi:10.1512/iumj.2001.50.1933.
3.
W.Cooperman, A near-optimal rate of periodic homogenization for convex Hamilton–Jacobi equations, Arch. Ration. Mech. Anal.245(2) (2022), 809–817. doi:10.1007/s00205-022-01797-x.
4.
M.Einseidler and T.Ward, Functional Analysis, Spectral Theory, and Applications, Graduate Texts in Mathematics, Vol. 276, Springer International Publishing AG, 2017.
5.
L.C.Evans and D.Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics II, Arch. Ration. Mech. Anal.161(1) (2002), 271–305. doi:10.1007/s002050100181.
6.
W.Jing, H.V.Tran and Y.Yu, Effective fronts of polytope shapes, Minimax Theory Appl.5(2) (2020), 347–360.
7.
P.-L.Lions, G.Papanicolaou and S.R.S.Varadhan, Homogenization of Hamilton–Jacobi equations, 1987, unpublished work.
8.
H.Mitake, H.V.Tran and Y.Yu, Rate of convergence in periodic homogenization of Hamilton–Jacobi equations: The convex setting, Arch. Ration. Mech. Anal.233(2) (2019), 901–934. doi:10.1007/s00205-019-01371-y.
9.
H.V.Tran, Hamilton–Jacobi Equations: Theory and Applications, American Mathematical Society, Graduate Studies in Mathematics, Vol. 213, 2021.
10.
H.V.Tran and Y.Yu, Optimal convergence rate for periodic homogenization of convex Hamilton–Jacobi equations, 2021, arXiv:2112.06896.
11.
S.N.T.Tu, Rate of convergence for periodic homogenization of convex Hamilton–Jacobi equations in one dimension, Asymptotic Analysis121(2) (2021), 171–194. doi:10.3233/ASY-201599.
