### Abstract

Uniform estimate for the solutions of elliptic equations with high-contrast conductivities in R^{n} is concerned. The space domain consists of a periodic connected sub-region and a periodic disconnected matrix block subset. The elliptic equations have fast diffusion in the connected sub-region and slow diffusion in the disconnected subset. Suppose ε∈(0, 1] is the diameter of each matrix block and ω^{2}∈(0, 1] is the conductivity ratio of the disconnected matrix block subset to the connected sub-region. It is proved that the W^{1,p} norm of the elliptic solutions in the connected sub-region is bounded uniformly in ε, ω when ε≤ω, the L^{p} norm of the elliptic solutions in the whole space is bounded uniformly in ε, ω the W^{1,p} norm of the elliptic solutions in perforated domains is bounded uniformly in ε. However, the L^{p} norm of the second order derivatives of the solutions in the connected sub-region may not be bounded uniformly in ε, ω.

Original language | English |
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Pages (from-to) | 925-966 |

Number of pages | 42 |

Journal | Journal of Differential Equations |

Volume | 261 |

Issue number | 2 |

DOIs | |

State | Published - 15 Jul 2016 |

### Keywords

- Duality argument
- Embedding theory
- High-contrast conductivity
- Potentials