Hamilton-Jacobi equations with partial gradient and application to homogenization

O. Alvarez*, H. Ishii

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

16 Citations (Scopus)


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
Issue number5-6
Publication statusPublished - 2001
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)
  • Analysis
  • Applied Mathematics


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