论文标题
椭圆运算符在较低边界的域上的Carleson扰动
Carleson perturbations of elliptic operators on domains with low dimensional boundaries
论文作者
论文摘要
我们证明了Fefferman，Kenig和Pipher形成差异问题的扰动结果的类似物，对于David，Feneuil和Mayboroda的退化椭圆运营商而言，这些问题是为了研究与该集合的数量和分析性能的元素和分析性能，其合成型的数量和分析性能高于1美元。这些运算符的形式为$ - \ text {div} a \ nabla $，其中$ a $是一种加权椭圆矩阵，旨在以一种允许滋养椭圆理论的方式来权衡到达高共二维边界的距离。当这个边界是$ \ alhfors-david以$ \ mathbb r^n $设置为$ d \ in [1，n-1）$和$ n \ geq3 $时，我们证明，在$ a _ {\ infty} $中，谐波度量的成员在Carleson Materion carleson Materive in the $ a _ {\ inftty} $中保存了$ clyciention $ coefferiencation $ coefferients $ n n y r^。在这些扰动下（可能有不同的$ p $），Dirichlet问题也稳定。如果Carleson测量扰动很小，我们将在同一$ l^p $空间中建立Dirichlet问题的解决性。我们的结果的推杆之一以及戴维（David），恩格斯坦（Engelstein）和梅博罗（Mayboroda）的先前结果是，在[1，n-2）$，$ n \ geq3 $中$ d $ -Adr边界$γ$，$ d \ in [1，n-2）$，$ n \ geq3 $，在上面描述的形式的一个偏爱量子的否定性均与$ d-$ d-dimens $ d-dimens auss n of the n harmocons所描述的形式相关。
We prove an analogue of a perturbation result for the Dirichlet problem of divergence form elliptic operators by Fefferman, Kenig and Pipher, for the degenerate elliptic operators of David, Feneuil and Mayboroda, which were developed to study geometric and analytic properties of sets with boundaries whose co-dimension is higher than $1$. These operators are of the form $-\text{div} A\nabla$, where $A$ is a weighted elliptic matrix crafted to weigh the distance to the high co-dimension boundary in a way that allows for the nourishment of an elliptic theory. When this boundary is a $d-$Alhfors-David regular set in $\mathbb R^n$ with $d\in[1,n-1)$ and $n\geq3$, we prove that the membership of the harmonic measure in $A_{\infty}$ is preserved under Carleson measure perturbations of the matrix of coefficients, yielding in turn that the $L^p-$solvability of the Dirichlet problem is also stable under these perturbations (with possibly different $p$). If the Carleson measure perturbations are suitably small, we establish solvability of the Dirichlet problem in the same $L^p$ space. One of the corollaries of our results together with a previous result of David, Engelstein and Mayboroda, is that, given any $d$-ADR boundary $Γ$ with $d\in[1,n-2)$, $n\geq3$, there is a family of degenerate operators of the form described above whose harmonic measure is absolutely continuous with respect to the $d-$dimensional Hausdorff measure on $Γ$.
