Hamilton-Jacobi equations with partial gradient and application to homogenization

O. Alvarez, H. Ishii

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

The paper proves that the Dirichlet problem for the first-order Hamilton-Jacobi equation in an open subset of ℝn H(x, u, Dx′u) = 0 in Ω, u = g on ∂Ω, where Dx′u is the partial gradient of the scalar function u with respect to the first n′ variables (n′ ≤ n), has a viscosity solution which is unique a.e. When applied to the periodic homogenization of Hamilton-Jacobi equations in a perforated set, the result yields the a.e. convergence of the solutions of the problem at scale ε as ε → 0 to the solution of the homogenized Hamilton-Jacobi equation.

Original languageEnglish
Pages (from-to)983-1002
Number of pages20
JournalCommunications in Partial Differential Equations
Volume26
Issue number5-6
Publication statusPublished - 2001
Externally publishedYes

Fingerprint

Hamilton-Jacobi Equation
Homogenization
Gradient
Partial
Periodic Homogenization
Almost Everywhere Convergence
Viscosity Solutions
Viscosity
Dirichlet Problem
Scalar
First-order
Subset

ASJC Scopus subject areas

  • Mathematics(all)
  • Analysis
  • Applied Mathematics

Cite this

Hamilton-Jacobi equations with partial gradient and application to homogenization. / Alvarez, O.; Ishii, H.

In: Communications in Partial Differential Equations, Vol. 26, No. 5-6, 2001, p. 983-1002.

Research output: Contribution to journalArticle

@article{98863b331f5348d9b20adf95324caba0,
title = "Hamilton-Jacobi equations with partial gradient and application to homogenization",
abstract = "The paper proves that the Dirichlet problem for the first-order Hamilton-Jacobi equation in an open subset of ℝn H(x, u, Dx′u) = 0 in Ω, u = g on ∂Ω, where Dx′u is the partial gradient of the scalar function u with respect to the first n′ variables (n′ ≤ n), has a viscosity solution which is unique a.e. When applied to the periodic homogenization of Hamilton-Jacobi equations in a perforated set, the result yields the a.e. convergence of the solutions of the problem at scale ε as ε → 0 to the solution of the homogenized Hamilton-Jacobi equation.",
author = "O. Alvarez and H. Ishii",
year = "2001",
language = "English",
volume = "26",
pages = "983--1002",
journal = "Communications in Partial Differential Equations",
issn = "0360-5302",
publisher = "Taylor and Francis Ltd.",
number = "5-6",

}

TY - JOUR

T1 - Hamilton-Jacobi equations with partial gradient and application to homogenization

AU - Alvarez, O.

AU - Ishii, H.

PY - 2001

Y1 - 2001

N2 - The paper proves that the Dirichlet problem for the first-order Hamilton-Jacobi equation in an open subset of ℝn H(x, u, Dx′u) = 0 in Ω, u = g on ∂Ω, where Dx′u is the partial gradient of the scalar function u with respect to the first n′ variables (n′ ≤ n), has a viscosity solution which is unique a.e. When applied to the periodic homogenization of Hamilton-Jacobi equations in a perforated set, the result yields the a.e. convergence of the solutions of the problem at scale ε as ε → 0 to the solution of the homogenized Hamilton-Jacobi equation.

AB - The paper proves that the Dirichlet problem for the first-order Hamilton-Jacobi equation in an open subset of ℝn H(x, u, Dx′u) = 0 in Ω, u = g on ∂Ω, where Dx′u is the partial gradient of the scalar function u with respect to the first n′ variables (n′ ≤ n), has a viscosity solution which is unique a.e. When applied to the periodic homogenization of Hamilton-Jacobi equations in a perforated set, the result yields the a.e. convergence of the solutions of the problem at scale ε as ε → 0 to the solution of the homogenized Hamilton-Jacobi equation.

UR - http://www.scopus.com/inward/record.url?scp=1642471469&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1642471469&partnerID=8YFLogxK

M3 - Article

VL - 26

SP - 983

EP - 1002

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

SN - 0360-5302

IS - 5-6

ER -