Carleson perturbations of elliptic operators on domains with low dimensional boundaries

TL;DR

This paper extends perturbation results for elliptic operators to those with boundaries of higher co-dimension, showing stability of harmonic measure and Dirichlet problem solvability under Carleson measure perturbations.

Contribution

It develops a new perturbation theory for degenerate elliptic operators on low-dimensional boundaries, establishing stability of harmonic measure and Dirichlet problem solvability.

Findings

Harmonic measure remains in A_infinity class under Carleson perturbations.

Dirichlet problem solvability in L^p spaces is stable under small perturbations.

Existence of degenerate operators with harmonic measure absolutely continuous to boundary measure.

Abstract

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 $\Gamma$ 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 $\Gamma$.