## Abstract

Let L_{ε} = -div(A(x/ε)∇), ε > 0 be a family of second order elliptic operators with real, symmetric coefficients on a bounded Lipschitz domain Ω in ℝ^{n}, subject to the Dirichlet boundary condition. Assuming that A(x) is periodic and belongs to VMO, we show that there exists δ > 0 independent of ε such that Riesz transforms ∇(L_{ε})^{-1/2} are uniformly bounded on L^{P}(Ω), where 1 < p < 3 + δ if n ≥ 3, and 1 < p < 4 + δ if n = 2. The ranges of p's are sharp. In the case of C ^{1} domains, we establish the uniform L^{P} boundedness of ∇(L_{ε})^{-1/2} for 1 < p < ∞ and n ≥ 2. As a consequence, we obtain the uniform W^{1,p} estimates for the elliptic homogenization problem L_{ε}u_{ε} = div f in Ω, u_{ε} = 0 on ∂Ω.

Original language | English |
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Pages (from-to) | 2283-2298 |

Number of pages | 16 |

Journal | Indiana University Mathematics Journal |

Volume | 57 |

Issue number | 5 |

DOIs | |

State | Published - 2008 |

## Keywords

- Homogenization
- Lipschitz domain
- Riesz transform

## ASJC Scopus subject areas

- General Mathematics

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