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 $-\text{div}A\mathrm{\nabla }$, 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 PDE theory. When this boundary is a d-Alhfors-David regular set in ${\mathbb{R}}^n$ with $d\in [1,n-1)$ and $n\ge 3$, 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 $d\in [1,n-2)$, $n\ge 3$, 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.

我们证明了Fefferman，Kenig和Pipher的椭圆形算子发散的Dirichlet问题的Dirichlet问题，David，Feneuil和Mayboroda的退化椭圆算子的摄动结果的类似物，它们被开发用于研究具有边界的集合的几何和解析性质共维数大于1。这些运算符的形式为 $-\text{div}\mathrm{一种}\mathrm{\nabla }$，其中A是加权椭圆矩阵，其经过精心设计以权衡到高维数边界的距离，以允许滋养椭圆PDE理论。当此边界是d -Alhfors-David正则集时${\mathbb{[R}}^{ñ}$ 与 $d\in [\mathrm{1个}，ñ-\mathrm{1个}）$ 和 $ñ\ge 3$，我们证明了谐波度量的成员 ${\mathrm{一种}}_{\infty }$ 在Carleson度量系数矩阵的摄动下得以保留，从而得出 ${\mathrm{大号}}^p$Dirichlet问题的可解性在这些扰动下也很稳定（p可能不同）。如果Carleson测度的摄动适当小，我们将在同一条件下建立Dirichlet问题的可解性${\mathrm{大号}}^p$空间。我们的结果与之前的David，Engelstein和Mayboroda的结果的推论之一是，给定任何d -ADR边界Γ$d\in [\mathrm{1个}，ñ-2）$， $ñ\ge 3$，存在上述形式的简并算子族，其谐波度量相对于Γ的d维Hausdorff度量绝对连续。