### Abstract

The paper proves that the Dirichlet problem for the first-order Hamilton-Jacobi equation in an open subset of ℝ^{n} H(x, u, D_{x′}u) = 0 in Ω, u = g on ∂Ω, where D_{x′}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 language | English |
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Pages (from-to) | 983-1002 |

Number of pages | 20 |

Journal | Communications in Partial Differential Equations |

Volume | 26 |

Issue number | 5-6 |

Publication status | Published - 2001 |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Applied Mathematics

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## Cite this

Alvarez, O., & Ishii, H. (2001). Hamilton-Jacobi equations with partial gradient and application to homogenization.

*Communications in Partial Differential Equations*,*26*(5-6), 983-1002.