论文标题
满足卡莱森条件的椭圆运营商的边界价值问题
Boundary value problems for elliptic operators satisfying Carleson condition
论文作者
论文摘要
在本文中，我们以简洁的形式介绍了最新结果，并提供了证明性的证明，关于$ l^p $ dirichlet的可溶性，在Lipschitz域上的标量椭圆方程的规律性和Neumann问题，其系数满足了各种Carleson条件。更准确地说，使用$ l = \ mbox {div}（a \ nabla）$，我们假设矩阵$ a $是椭圆形的，并且以（$ | \ nabla a（x）| \ lyssim \ lyssim \ mbox \ mbox \ mbox {dist}（x，\ partialpial gontial gomance）（x，\ nabla a（x）^，\ nabla a（x） a |（x）^2 \ mbox {dist}（x，x，\partialΩ）\，dx $）或$ \ mbox {dist}（x，x，\partialΩ）^{ - 1} \ left（\ mbox {\ mbox {osc {osc} _ {b（x，Δ（x），x，Δ（x）} a \ resule 我们提出了两种类型的结果，第一个是所谓的“小卡勒森”情况，在给定的$ 1 <p <\ infty $中，我们证明了这三个被认为是该系数的Carleson Norm和所考虑域的Lipschitz常数的三个考虑边界价值问题的可溶性。第二种结果（“大型Carleson”）放宽了对任何Lipschitz结构域的约束，并假设系数的Carleson Norm只是界定。在这种情况下，我们在$（1，\ infty）$的子间隔中有$ l^p $ solvicality的$ p $范围。 在本文的结尾，我们简要概述了有关Lipschitz以外的域（例如统一域或和弦 - ARC域）的最新结果。
In this paper we present in concise form recent results, with illustrative proofs, on solvability of the $L^p$ Dirichlet, Regularity and Neumann problems for scalar elliptic equations on Lipschitz domains with coefficients satisfying a variety of Carleson conditions. More precisely, with $L=\mbox{div}(A\nabla)$, we assume the matrix $A$ is elliptic and satisfies a natural Carleson condition either in the form that ($|\nabla A(X)|\lesssim \mbox{dist}(X,\partialΩ)^{-1}$ and $|\nabla A|(X)^2\mbox{dist}(X,\partialΩ)\,dX$) or $\mbox{dist}(X,\partialΩ)^{-1}\left(\mbox{osc}_{B(X,δ(X)/2)}A\right)^2\,dX$ is a Carleson measure. We present two types of results, the first is the so-called "small Carleson" case where, for a given $1<p<\infty$, we prove solvability of the three considered boundary value problems under assumption the Carleson norm of the coefficients and the Lipschitz constant of the considered domain is sufficiently small. The second type of results ("large Carleson") relaxes the constraints to any Lipschitz domain and to the assumption that the Carleson norm of the coefficients is merely bounded. In this case we have $L^p$ solvability for a range of $p$'s in a subinterval of $(1,\infty)$. At the end of the paper we give a brief overview of recent results on domains beyond Lipschitz such as uniform domains or chord-arc domains.
