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 $L_0u=-\mathrm{div}(A_0\nabla u)$, $Lu=-\mathrm{div}(A\nabla u)$ 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 $RH_q$-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, $\omega_L\in RH_q(\omega_{L_0})$ 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.