## Abstract

This chapter is devoted to the study of uniform boundary regularity estimates for the Dirichlet problem [Formula presented.] where Lε =div(A(x/ε)∇). Assuming that the coefficient matrix A = A(y) is elliptic, periodic, and belongs to VMO(ℝ^{d}), we establish uniform boundary Hölder and W^{1, p} estimates in C^{1} domains Ω. We also prove uniform boundary Lipschitz estimates in C^{1, α} domains under the assumption that A is elliptic, periodic, and Hölder continuous. As in the previous chapter for interior estimates, boundary Hölder and Lipschitz estimates are proved by a compactness method. The boundaryW^{1, p} estimates are obtained by combining the boundary Hölder estimates with the interior W^{1, p} estimates, via the real-variable method introduced in Section 4.2.

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

Title of host publication | Operator Theory |

Subtitle of host publication | Advances and Applications |

Pages | 99-134 |

Number of pages | 36 |

DOIs | |

State | Published - 2018 |

### Publication series

Name | Operator Theory: Advances and Applications |
---|---|

Volume | 269 |

ISSN (Print) | 0255-0156 |

ISSN (Electronic) | 2296-4878 |

### Bibliographical note

Publisher Copyright:© Springer Nature Switzerland AG 2018.

## ASJC Scopus subject areas

- Analysis