## Abstract

Convergence for the solutions of elliptic equations in periodic perforated domains is concerned. Let ε denote the size ratio of the holes of a periodic perforated domain to the whole domain. It is known that, by energy method, the gradient of the solutions of elliptic equations is bounded uniformly in ε in L^{2} space. Also, when ε approaches 0, the elliptic solutions converge to a solution of some simple homogenized elliptic equation. In this work, above results are extended to general W^{1,p} space for p>1. More precisely, a uniform W^{1,p} estimate in ε for p∈(1, ∞] and a W^{1,p} convergence result for p∈(nn-2,∞] for the elliptic solutions in periodic perforated domains are derived. Here n is the dimension of the space domain. One also notes that the L^{p} norm of the second order derivatives of the elliptic solutions in general cannot be bounded uniformly in ε.

Original language | English |
---|---|

Pages (from-to) | 1734-1783 |

Number of pages | 50 |

Journal | Journal of Differential Equations |

Volume | 255 |

Issue number | 7 |

DOIs | |

State | Published - 1 Oct 2013 |

## Keywords

- Homogenized elliptic equation
- Periodic perforated domain