论文标题
泊松 - 迪里奇莱特问题的解决性与$ l^{p'} $中的室内数据及其在$ l^{p} $的应用
Solvability of the Poisson-Dirichlet problem with interior data in $L^{p'}$-Carleson spaces and its applications to the $L^{p}$-regularity problem
论文作者
论文摘要
我们证明，$ l^{p'} $ - 椭圆运算符的均质dirichlet问题的溶解度$ l = - \ operatoTorname {div} a \ nabla $具有真实且仅限的系数等于$ l^{p'} $ - 可溶解度$ $ $ lw = h.假设$ l^{p'} $估算在非区域最大功能上，假设$ \ permatatorName {dist}（\ cdot，\ cdot，\ partialω）h $和$ f $均为某些$ l^{p'} $ - carleson-type-type $ carleson-type $ subset+subsset+s subsset+s r \ s r \ s r \ s r \ s r \ s r \ \ s r \ \ s r \ \ s SubsSse $ n \ geq2 $，满足开瓶器状况，并具有$ n $ -Ahlfors的常规边界。 In turn, we use this result to show that, in a bounded domain with uniformly $n$-rectifiable boundary that satisfies the corkscrew condition, $L^{p'}$-solvability of the homogeneous Dirichlet problem for an operator $L=-\operatorname{div} A\nabla$ satisfying the Dahlberg-Kenig-Pipher condition (of arbitrarily large constant) implies $ l^p $ - 置换问题的可解决性问题$ l^*= - \ operatatorName {div} a^t \ nabla $，其中$ 1/p+1/p'= 1 $和$ a^t $是$ a $的threspose矩阵矩阵。尽管$ l^{p'} $ - 自2001年以来，这些操作员在Lipschitz域中闻名了这些运营商的$ l^{p'} $，但Dahlberg-kenig-pipher操作员的这一结果即使是单位球，即使$ω$是单位球的新结果。 进一步的新颖应用程序包括i）对绿色功能及其在粗糙领域中的梯度的新的本地估计，ii）$ l^{p} $ - ``poisson-regularity问题''的本地$ t1 $ -type定理''本身等于$ l^{p'p'} $ - 溶解的本身，以概述本地的概述。函数和iii）在有限的粗糙域上的对称操作员$ l $的本征functions（及其梯度）的新$ l^p $估计值。
We prove that the $L^{p'}$-solvability of the homogeneous Dirichlet problem for an elliptic operator $L=-\operatorname{div}A\nabla$ with real and merely bounded coefficients is equivalent to the $L^{p'}$-solvability of the Poisson Dirichlet problem $Lw=H-\operatorname{div} F$, which is defined in terms of an $L^{p'}$ estimate on the non-tangential maximal function, assuming that $\operatorname{dist}(\cdot, \partial Ω) H$ and $F$ lie in certain $L^{p'}$-Carleson-type spaces, and that the domain $Ω\subset\mathbb R^{n+1}$, $n\geq2$, satisfies the corkscrew condition and has $n$-Ahlfors regular boundary. In turn, we use this result to show that, in a bounded domain with uniformly $n$-rectifiable boundary that satisfies the corkscrew condition, $L^{p'}$-solvability of the homogeneous Dirichlet problem for an operator $L=-\operatorname{div} A\nabla$ satisfying the Dahlberg-Kenig-Pipher condition (of arbitrarily large constant) implies solvability of the $L^p$-regularity problem for the adjoint operator $L^*=-\operatorname{div} A^T \nabla$, where $1/p+1/p'=1$ and $A^T$ is the transpose matrix of $A$. This result for Dahlberg-Kenig-Pipher operators is new even if $Ω$ is the unit ball, despite the fact that the $L^{p'}$-solvability of the Dirichlet problem for these operators in Lipschitz domains has been known since 2001. Further novel applications include i) new local estimates for the Green's function and its gradient in rough domains, ii) a local $T1$-type theorem for the $L^{p}$-solvability of the ``Poisson-Regularity problem'', itself equivalent to the $L^{p'}$-solvability of the homogeneous Dirichlet problem, in terms of certain gradient estimates for local landscape functions, and iii) new $L^p$ estimates for the eigenfunctions (and their gradients) of symmetric operators $L$ on bounded rough domains.
