Regularity and Neumann problems for operators with real coefficients satisfying Carleson condition
| Autor: | Martin Dindoš, Steve Hofmann, Jill Pipher |
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| Jazyk: | angličtina |
| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Dindos, M, Hofmann, S & Pipher, J 2023, ' Regularity and Neumann Problems for Operators with Real Coefficients Satisfying Carleson Conditions ', Journal of functional analysis, vol. 285, no. 6, 110024 . https://doi.org/10.1016/j.jfa.2023.110024 |
| DOI: | 10.1016/j.jfa.2023.110024 |
| Popis: | In this paper, we continue the study of a class of second order elliptic operators of the form $\mathcal L=\mbox{div}(A\nabla\cdot)$ in a domain above a Lipschitz graph in $\mathbb R^n,$ 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 $1 < q < \infty$. 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 $11$ is the number such that the $L^q$ Dirichlet problem for the adjoint operator $\mathcal L^*$ is solvable for all $q>q_*$. Additionally when $n=2$, there exists $p_{neum}>1$ such that for all $11$ is the number such that the $L^q$ Dirichlet problem for the operator $\mathcal L_1=\mbox{div}(A_1\nabla\cdot)$ with matrix $A_1=A/\det{A}$ is solvable for all $q>q^*$. Comment: 27 pages. V2 has an updated and shortened argument |
| Databáze: | OpenAIRE |
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