## Abstract

We consider divergence form elliptic operators of the form L=-divA(x)▼, defined in R^{n+1}={(x,t)εR^{n}×R}, n≥2, where the L^{∞} coefficient matrix A is (n+1)×(n+1), uniformly elliptic, complex and t-independent. We show that for such operators, boundedness and invertibility of the corresponding layer potential operators on L^{2}(R^{n})=L^{2}(∂R_{+}^{n+1}), is stable under complex, L^{∞} perturbations of the coefficient matrix. Using a variant of the Tb Theorem, we also prove that the layer potentials are bounded and invertible on L^{2}(R^{n}) whenever A(x) is real and symmetric (and thus, by our stability result, also when A is complex, ||A-A^{0}||_{∞} is small enough and A^{0} is real, symmetric, L^{∞} and elliptic). In particular, we establish solvability of the Dirichlet and Neumann (and Regularity) problems, with L^{2} (resp. L _{1}^{2}) data, for small complex perturbations of a real symmetric matrix. Previously, L^{2} solvability results for complex (or even real but non-symmetric) coefficients were known to hold only for perturbations of constant matrices (and then only for the Dirichlet problem), or in the special case that the coefficients A_{j},n+1=0=A_{n+1,j}, 1≤j≤n, which corresponds to the Kato square root problem.

Original language | English |
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Pages (from-to) | 4533-4606 |

Number of pages | 74 |

Journal | Advances in Mathematics |

Volume | 226 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2011 Mar 20 |

### Bibliographical note

Funding Information:* Corresponding author. E-mail addresses: maria.alfonseca@ndsu.edu (M.A. Alfonseca), pascal.auscher@math.u-psud.fr (P. Auscher), andreas.axelsson@liu.se (A. Axelsson), hofmanns@missouri.edu (S. Hofmann), kimseick@yonsei.ac.kr (S. Kim). 1 S. Hofmann was supported by the National Science Foundation.

## All Science Journal Classification (ASJC) codes

- Mathematics(all)