Regularity and Neumann problems for operators with real coefficients satisfying Carleson conditions
Regularity and Neumann problems for operators with real coefficients satisfying Carleson conditions
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DOI：
10.1016/j.jfa.2023.110024
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发表时间：
2022-07
影响因子：
1.7
通讯作者：
M. Dindoš;S. Hofmann;J. Pipher
中科院分区：
文献类型：
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作者：
M. Dindoš;S. Hofmann;J. Pipher
本文继续研究了Rn中Lipschitz图上区域上的一类二阶椭圆算子L= div（A_div），其中矩阵A的系数满足Carleson测度条件，表示为Whitney球上振动的条件.对于这类算子，已知（自2001年以来）L q Dirichlet问题对于某些1< q<∞是可解的。进一步的研究完全解决了当振动的Carleson测度范数足够小时，Lipschitz域上Dirichlet，Regularity，Neumann问题的Lq可解性的范围.我们证明了存在p re g> 1使得对所有的1< p< p re g，算子L= div（A_p）的Lp正则性问题是可解的.此外，1 p r e g+ 1 q = 1，其中q> 1是使得伴随算子L的L q狄利克雷问题对于所有q> q都可解的数字。另外，当n= 2时，存在p n e um> 1，使得对于所有1< p< p n e um，算子L= div（A_n）的Lp Neumann问题是可解的。此外，1 p re g+ 1 q = 1，其中q> 1是使得算子L1 = div（A1 <$$>）的Lq Dirichlet问题对所有q> q都可解的数，其中矩阵A1 = A/det <$A.
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 r e g> 1 such that for all 1< p< p r e g the L p Regularity problem for the operator L= div (A∇⋅) is solvable. Furthermore 1 p r e g+ 1 q⁎= 1 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 n e u m> 1 such that for all 1< p< p n e u m the L p Neumann problem for the operator L= div (A∇⋅) is solvable. Furthermore 1 p r e g+ 1 q⁎= 1 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/det A is solvable for all q> q⁎.