### 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 |
---|---|

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 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Applied Mathematics

### Cite this

*Communications in Partial Differential Equations*,

*26*(5-6), 983-1002.

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

Research output: Contribution to journal › Article

*Communications in Partial Differential Equations*, vol. 26, no. 5-6, pp. 983-1002.

}

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 -