Let $u^\varepsilon$ 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 $\mathcal{O}(\varepsilon)$ of $u^\varepsilon \rightarrow u$ as $\varepsilon \rightarrow 0^+$ 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 $n = 1$.
We study homogenization for a class of stationnary Hamilton-Jacobi equations in which the Hamiltonian is obtained by perturbing near the origin an otherwise periodic Hamiltonian. We prove that the limiting problem consists of a Hamilton-Jacobi equation outside the origin, with the same effective Hamiltonian as in periodic homogenization, supplemented at the origin with an effective Dirichlet condition that keeps track of the perturbation. Various comments and extensions are discussed.
Multiscale Homogenizations for first-order Hamilton-Jacobi-Bellman Equations
研究了几类一阶Hamilton-Jaconi-Bellman方程的拟周期均匀化问题。拟周期均匀化的胞腔问题满足非共振条件，在此条件下相应的确定性系统是遍历的。同样的类方程的概周期均匀化也解决了，作为一个序列的准周期均匀化的限制。这里，概周期性是H的意义。波尔这一结果已被一些作者引用，例如：H. Ishii，“Hamilton-Jacobi方程的几乎周期均匀化”，在Int.方程，第1卷，柏林，1999年，《世界科学》，River Edge，NJ 2000年，600-605;和P. - L.狮子和体育Souganstan，“Corrector for the Homogenizations of Hamilton-Jacobi Equations in the stationary ergodic setting”，Comm. Pure Appl. Math. LVI，（2003），pp. 1501-1524.
In this paper we provide a rate of convergence for periodic homogenization of Hamilton-Jacobi-Bellman equations with nonlocal diffusion. The result is based on the regularity of the associated effective problem, where the convexity plays a crucial role. Such regularity estimates are possible from the available literature once we provide a representation formula for the effective Hamiltonian, a result that has an independent interest.
We consider a Hamilton-Jacobi equation where the Hamiltonian is periodic in space and coercive and convex in momentum. Combining the representation formula from optimal control theory and a theorem of Alexander, originally proved in the context of first-passage percolation, we find a rate of homogenization which is within a log-factor of optimal and holds in all dimensions.
Here, we consider periodic homogenization for time-fractional Hamilton--Jacobi equations. By using the perturbed test function method, we establish the convergence, and give estimates on a rate of convergence. A main difficulty is the incompatibility between the function used in the doubling variable method, and the non-locality of the Caputo derivative. Our approach is to provide a lemma to prove the rate of convergence without the doubling variable method with respect to the time variable, which is a key ingredient.
We study the rate of convergence in periodic homogenization for convex Hamilton--Jacobi equations with multiscales, where the Hamiltonian $H=H(x, y, p): \mathbb{R}^n \times \mathbb{T}^n \times \mathbb{R}^n \to \mathbb{R }$ depends on both of the spatial variable and the oscillatory variable. In particular, we show that for the Cauchy problem, the rate of convergence is $O(\sqrtε)$ by optimal control formulas, scale separations and curve cutting techniques. We also show the rate $O(\sqrtε)$ of homogenization for the static problem based on the same idea. Additionally, we provide examples that i…
Capuzzo-Dolcetta and Ishii proved that the rate of periodic homogenization for coercive Hamilton-Jacobi equations is $O(\varepsilon^{1/3})$. We complement this result by constructing examples of coercive nonconvex Hamiltonians whose rate of periodic homogenization is $Ω(\varepsilon^{1/2})$.
Optimal convergence rates in periodic homogenization of nonconvex Hamilton--Jacobi equations
University of Maryland-College Park（马里兰大学帕克分校）；Rutgers University（罗格斯大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of California at Irvine（加州大学尔湾分校）
We study the optimal convergence rate for 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 [8], which means the optimal convergence rate is also $O(\varepsilon)$.
We study the homogenization of nonlinear, first-order equations with highly oscillatory mixing spatio-temporal dependence. It is shown in a variety of settings that the homogenized equations are stochastic Hamilton-Jacobi equations with deterministic, spatially homogenous Hamiltonians driven by white noise in time. The paper also contains proofs of some general regularity and path stability results for stochastic Hamilton-Jacobi equations, which are needed to prove some of the homogenization results and are of independent interest.
In this paper, we show that the rate of convergence in periodic homogenization of convex Hamilton-Jacobi equations is always $O(\varepsilon)$, which is optimal. This is a natural extension of a result concerning stable norms in metric geometry [4] that is essentially equivalent to the homogenization of convex static Hamilton-Jacobi equations. Another extremely interesting question in this direction is whether the $O(\varepsilon)$ rate holds in the nonconvex setting. We present a special nonconvex example with $O(\varepsilon)$ convergence rate, which relies on identifying the shape of the effec…
We study the homogenization of a Hamilton-Jacobi equation forced by rapidly oscillating noise that is colored in space and white in time. It is shown that the homogenized equation is deterministic, and, in general, the noise has an enhancement effect, for which we provide a quantitative estimate. As an application, we perform a noise sensitivity analysis for Hamilton-Jacobi equations forced by a noise term with small amplitude, and identify the scaling at which the macroscopic enhancement effect is felt. The results depend on new, probabilistic estimates for the large scale Hölder regularity o…
We prove homogenization for viscous Hamilton-Jacobi equations with a Hamiltonian of the form $G(p)+V(x,ω)$ for a wide class of stationary ergodic random media in one space dimension. The momentum part $G(p)$ of the Hamiltonian is a general (nonconvex) continuous function with superlinear growth at infinity, and the potential $V(x,ω)$ is bounded and Lipschitz continuous. The class of random media we consider is defined by an explicit hill and valley condition on the diffusivity-potential pair which is fulfilled as long as the environment is not ``rigid''.
