Almost periodic homogenization theory for Hamilton–Jacobi equations

2021 ◽  
pp. 163-176
Keyword(s):  
Homogenization Theory ◽  
Almost Periodic ◽  
Periodic Homogenization ◽  
Jacobi Equations

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

2021 ◽  
Vol 121 (2) ◽  
pp. 171-194
Author(s):  
Son N.T. Tu
Keyword(s):  
Rate Of Convergence ◽  
Classical Mechanics ◽  
One Dimension ◽  
Separable Potential ◽  
Periodic Homogenization ◽  
Periodic Functions ◽  
Effective Equation ◽  
Optimal Rate Of Convergence ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

Let u ε 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 u ε → 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.

scholarly journals Time Remotely Almost Periodic Viscosity Solutions of Hamilton-Jacobi Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Shilin Zhang ◽  
Daxiong Piao
Keyword(s):  
Viscosity Solutions ◽  
Existence And Uniqueness ◽  
Almost Periodic Functions ◽  
Periodic Case ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Jacobi Equations

We study some properties of the remotely almost periodic functions. This paper studies viscosity solutions of general Hamilton-Jacobi equations in the time remotely almost periodic case. Existence and uniqueness results are presented under usual hypotheses.

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