## Abstract

We consider divergence form elliptic equations Lu:=∇{dot operator}(A∇u)=0 in the half space R_{+}^{n+1}:={(x,t)∈R^{n}×(0,∞)}, whose coefficient matrix A is complex elliptic, bounded and measurable. In addition, we suppose that A satisfies some additional regularity in the direction transverse to the boundary, namely that the discrepancy A(x, t)-A(x, 0) satisfies a Carleson measure condition of Fefferman-Kenig-Pipher type, with small Carleson norm. Under these conditions, we establish a full range of boundedness results for double and single layer potentials in L^{p}, Hardy, Sobolev, BMO and Hölder spaces. Furthermore, we prove solvability of the Dirichlet problem for L, with data in L^{p}(R^{n}), BMO(R^{n}), and C^{α}(R^{n}), and solvability of the Neumann and Regularity problems, with data in the spaces L^{p}(R^{n})/H^{p}(R^{n}) and L_{1}^{p}(R^{n})/H^{1,p}(R^{n}) respectively, with the appropriate restrictions on indices, assuming invertibility of layer potentials for the t-independent operator L_{0}:=-∇{dot operator}(A({dot operator}, 0)∇).

Original language | English (US) |
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Pages (from-to) | 480-564 |

Number of pages | 85 |

Journal | Advances in Mathematics |

Volume | 270 |

DOIs | |

State | Published - Jan 2 2015 |

### Bibliographical note

Publisher Copyright:© 2014 Elsevier Inc.

## Keywords

- Boundary value problems
- Carleson measures
- Complex coefficients
- Elliptic operator
- Layer potentials

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