Perturbation of elliptic operators in 1-sided NTA domains satisfying the capacity density condition

Abstract

Let $\Omega \subset {\mathbb{R}}^{n+1}$, $n\ge 2$, be a 1-sided non-tangentially accessible domain (aka uniform domain), i.e., a set which satisfies the interior Corkscrew and Harnack chain conditions, respectively scale-invariant/quantitative versions of openness and path-connectedness. Assume that $\Omega$ satisfies the so-called capacity density condition. Let $L_{0}u=-\mathrm{div}(A_0\nabla u)$, $Lu=-\mathrm{div}(A\nabla u)$ be two real (non-necessarily symmetric) uniformly elliptic operators, and write ${\omega }_{L_0}$, ${\omega }_L$ for the associated elliptic measures. The goal of this program is to find sufficient conditions guaranteeing that ${\omega }_L$ satisfies an $A_{\infty }$-condition or a $R{H}_q$-condition with respect to ${\omega }_{L_0}$. We show that if the discrepancy of the two matrices satisfies a natural Carleson measure condition with respect to ${\omega }_{L_0}$, then ${\omega }_L\in A_{\infty }({\omega }_{L_0})$. Moreover, ${\omega }_L\in R{H}_q({\omega }_{L_0})$ for any given $1<q<\infty$ if the Carleson measure condition is assumed to hold with a sufficiently small constant. This extends previous work of Fefferman-Kenig-Pipher and Milakis-Pipher-Toro who considered Lipschitz and chord-arc domains. Here we go beyond as the capacity density condition is much weaker than the existence of exterior Corkscrew balls. The "large constant" case, where the discrepancy satisfies a Carleson measure condition, is new even for nice domains such as the unit ball, the upper half-space, or Lipschitz domains, and is obtained using the method of extrapolation of Carleson measure. Our domains do not have a nice surface measure: all the analysis is done with the underlying measure ${\omega }_{L_0}$. When particularized to Lipschitz, chord-arc, or 1-sided chord-arc domains, we recover previous results and extend some of them. Our arguments rely on the square function and non-tangential estimates proved in arXiv:2103.10046.

This paper is part of the earlier submission arXiv:1901.08261v2