Periodic homogenization theory for Hamilton–Jacobi equations

2021 ◽  
pp. 109-162
Keyword(s):  
Homogenization Theory ◽  
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.

Localization and multiplicity in the homogenization of nonlinear problems

2019 ◽  
Vol 9 (1) ◽  
pp. 292-304 ◽  
Author(s):  
Renata Bunoiu ◽  
Radu Precup
Keyword(s):  
Functional Analysis ◽  
Fixed Point Theory ◽  
Nonlinear Problems ◽  
Point Theory ◽  
Homogenization Theory ◽  
Infinitely Many Solutions ◽  
Significant Gain ◽  
Periodic Homogenization ◽  
Nonlinear Functional Analysis ◽  
Nonlinear Functional

Abstract We propose a method for the localization of solutions for a class of nonlinear problems arising in the homogenization theory. The method combines concepts and results from the linear theory of PDEs, linear periodic homogenization theory, and nonlinear functional analysis. Particularly, we use the Moser-Harnack inequality, arguments of fixed point theory and Ekeland's variational principle. A significant gain in the homogenization theory of nonlinear problems is that our method makes possible the emergence of finitely or infinitely many solutions.

scholarly journals Cauchy problem and periodic homogenization for nonlocal Hamilton–Jacobi equations with coercive gradient terms

2019 ◽  
Vol 150 (6) ◽  
pp. 3028-3059
Author(s):  
Martino Bardi ◽  
Annalisa Cesaroni ◽  
Erwin Topp
Keyword(s):  
Cauchy Problem ◽  
Uniform Convergence ◽  
Unique Solution ◽  
Comparison Principle ◽  
Nonlocal Operator ◽  
Periodic Homogenization ◽  
Effective Equation ◽  
Superlinear Growth ◽  
The Cauchy Problem ◽  
Jacobi Equations

AbstractThis paper deals with the periodic homogenization of nonlocal parabolic Hamilton–Jacobi equations with superlinear growth in the gradient terms. We show that the problem presents different features depending on the order of the nonlocal operator, giving rise to three different cell problems and effective operators. To prove the locally uniform convergence to the unique solution of the Cauchy problem for the effective equation we need a new comparison principle among viscosity semi-solutions of integrodifferential equations that can be of independent interest.

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