Effective Hamiltonian and homogenization of measurable eikonal equations
type: Item
instance of: scholarly article
title: Effective Hamiltonian and homogenization of measurable eikonal equations
zbMATH Open document ID: 1173.35014
author: Fabio Camilli
author: Antonio Siconolfi
published in: Archive for Rational Mechanics and Analysis
publication date: 2009-03-26T00:00:00Z
review text: The authors describe the homogenization of evolution Hamilton-Jacobi equations \(u_{\varepsilon t}+H(Du_{\varepsilon })=f(x/\varepsilon )\) posed in \(\mathbb R^{N}\times (0,+\infty)\). Here the potential \(f\) is supposed to be \(\mathbb Z^{N}\)-periodic, measurable and essentially bounded. The function \(H\) satisfies the usual conditions of convexity and coercivity. The solution \( u_{\varepsilon }\) starts from a uniformly continuous data \(u_{0}\) at \(t=0\). The main result of the paper proves that \(u_{\varepsilon }\) converges locally and uniformly in \(\mathbb R^{N}\times (0,+\infty )\) to the solution of \( u_{t}+\overline{H}(Du)=0\) posed in \(\mathbb R^{N}\times (0,+\infty )\), where \( \overline{H}\) is the unique value of the parameter \(a\) which makes the cell problem \(H(Du+p_{0})=f(x)+a\) solvable by a periodic function, for every \( p_{0}\). The authors first define an appropriate notion of viscosity subsolution, supersolution and solution for the original problem. They prove some properties of these solutions and prove the existence of a unique uniformly continuous solution \(u_{\varepsilon }\) of the original problem. They then build the notion of viscosity solution for the cell problem, which leads to the existence of a unique viscosity solution of this cell problem. The last part of the paper proves the homogenization result. This is done using appropriate subsolutions and supersolutions.
reviewed by: Alain Brillard
Mathematics Subject Classification ID: 35B27
Mathematics Subject Classification ID: 35F25
Mathematics Subject Classification ID: 49J45
zbMATH DE Number: 5533119
zbMATH Keywords: Hamilton-Jacobi equation
zbMATH Keywords: viscosity subsolution
zbMATH Keywords: viscosity supersolution
zbMATH Keywords: viscosity solution
zbMATH Keywords: cell problem
MaRDI profile type: Publication
full work available at URL: s00205-006-0001-0
OpenAlex ID: W1985777842
cites work: Singular perturbations of nonlinear degenerate parabolic pDEs: A general convergence result
cites work: Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations
cites work: Hamilton-Jacobi equations with measurable dependence on the state variable.
cites work: Time-Dependent Measurable Hamilton–Jacobi Equations
cites work: Periodic homogenization of Hamilton-Jacobi equations: Additive eigenvalues and variational formula
cites work: Periodic homogenisation of Hamilton–Jacobi equations: 2. Eikonal equations
cites work: LIP manifolds: from metric to Finslerian structure
cites work: Periodic homogenisation of certain fully nonlinear partial differential equations
cites work: PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians
cites work: Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting
DOI: 10.1007/S00205-006-0001-0
description: scientific article; zbMATH DE number 5533119