In this paper we study the rate of convergence u ε −→ u as ε −→ 0 + in periodic homogenization of Hamilton-Jacobi equations. Here u ε and u are viscosity solutions to the oscillatory Hamilton-Jacobi equationand its effective equationrespectively. Assuming that the initial data u 0 is bounded and Lipschitz continuous, we provide a simple proof to get the optimal rate O(ε) for a class of Hamiltonians including the classical mechanics one with separable potentialwhere a(·) : R −→ (0, +∞) is continuously differentiable, b(·) : R −→ (−∞, 0] is continuous and 1-periodic.2010 Mathematics Subject Classification. 35B40, 37J50, 49L25 .
We study the asymptotic behavior, as λ → 0 + , of the state-constraint Hamilton-Jacobi equationHere, Ω is a bounded domain of R n and φ(λ), r(λ) : (0, ∞) → (0, ∞) are continuous nondecreasing functions such that lim λ→0 + φ(λ) = lim λ→0 + r(λ) = 0. A similar problem on (1 + r(λ))Ω is also considered. Surprisingly, we are able to obtain both convergence results and non-convergence results in this setting. Moreover, we provide a very first result on the asymptotic expansion of the additive eigenvalue of H in (1 ± r(λ))Ω as λ → 0 + .
I ntroductionLet φ(λ), r(λ) : (0, ∞) → (0, ∞) be continuous nondecreasing functions such that lim λ→0 + φ(λ) = lim λ→0 + r(λ) = 0. We study the asymptotic behavior, as the discount factor φ(λ) goes to 0, of the viscosity solutions to the following state-constraint Hamilton-Jacobi equationHere, Ω is a bounded domain of R n . For simplicity, we will write Ω λ = (1 − r(λ))Ω and Ω λ = (1 + r(λ))Ω for λ > 0. A similar problem for Ω λ will also be considered. Roughly speaking, along some subsequence λ j → 0 + , we obtain the limiting equation as a state-constraint ergodic problem: H(x, Du(x)) c 0 in Ω, H(x, Du(x)) c 0 on Ω.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.