Rate of convergence for periodic homogenization of convex Hamilton–Jacobi equations in one dimension

Article type: Research Article

Authors: Tu, Son N.T.^{; *}

Affiliations: Department of Mathematics, University Wisconsin-Madison, WI, USA. E-mail: thaison@math.wisc.com

Correspondence: [*] Corresponding author. E-mail: thaison@math.wisc.com.

Abstract: Let uε and u be viscosity solutions of the oscillatory Hamilton–Jacobi equation and its corresponding effective equation. Given bounded, Lipschitz initial data, we present a simple proof to obtain the optimal rate of convergence O(ε) of uε→u as ε→0+ for a large class of convex Hamiltonians H(x,y,p) in one dimension. This class includes the Hamiltonians from classical mechanics with separable potential. The proof makes use of optimal control theory and a quantitative version of the ergodic theorem for periodic functions in dimension n=1.

Keywords: Cell problems, periodic homogenization, first order Hamilton–Jacobi equations, rate of convergence, viscosity solutions

DOI: 10.3233/ASY-201599

Journal: Asymptotic Analysis, vol. 121, no. 2, pp. 171-194, 2021