## Abstract / Description of output

In this paper, we continue the study of a class of second order elliptic operators of the form L=div(A∇⋅) in a domain above a Lipschitz graph in R
^{n}, where the coefficients of the matrix A satisfy a Carleson measure condition, expressed as a condition on the oscillation on Whitney balls. For this class of operators, it is known (since 2001) that the L
^{q} Dirichlet problem is solvable for some 1<q<∞. Moreover, further studies completely resolved the range of L
^{q} solvability of the Dirichlet, Regularity, Neumann problems in Lipschitz domains, when the Carleson measure norm of the oscillation is sufficiently small. We show that there exists p
_{reg}>1 such that for all 1<p<p
_{reg} the L
^{p} Regularity problem for the operator L=div(A∇⋅) is solvable. Furthermore [Formula presented] where q
_{⁎}>1 is the number such that the L
^{q} Dirichlet problem for the adjoint operator L
^{⁎} is solvable for all q>q
_{⁎}. Additionally when n=2, there exists p
_{neum}>1 such that for all 1<p<p
_{neum} the L
^{p} Neumann problem for the operator L=div(A∇⋅) is solvable. Furthermore [Formula presented] where q
^{⁎}>1 is the number such that the L
^{q} Dirichlet problem for the operator L
_{1}=div(A
_{1}∇⋅) with matrix A
_{1}=A/detA is solvable for all q>q
^{⁎}.

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

Article number | 110024 |

Journal | Journal of functional analysis |

Volume | 285 |

Issue number | 6 |

Early online date | 19 May 2023 |

DOIs | |

Publication status | Published - 15 Sept 2023 |

## Keywords / Materials (for Non-textual outputs)

- Boundary value problem
- Carleson measure
- Elliptic second order PDE
- Regularity problem