Akman, Murat and Hofmann, Steve and Martell, José María and Toro, Tatiana (2019) 'Perturbation of elliptic operators in 1sided NTA domains satisfying the capacity density condition.' Advances in Calculus of Variations. ISSN 18648258 (Submitted)

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Abstract
Let $\Omega\subset\mathbb{R}^{n+1}$, $n\ge 2$, be a 1sided nontangentially accessible domain (aka uniform domain), i.e., a set which satisfies the interior Corkscrew and Harnack chain conditions, respectively scaleinvariant/quantitative versions of openness and pathconnectedness. Assume that $\Omega$ satisfies the socalled capacity density condition. Let $L_0u=\mathrm{div}(A_0\nabla u)$, $Lu=\mathrm{div}(A\nabla u)$ be two real (nonnecessarily 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 FeffermanKenigPipher and MilakisPipherToro who considered Lipschitz and chordarc 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 halfspace, 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, chordarc, or 1sided chordarc domains, we recover previous results and extend some of them. Our arguments rely on the square function and nontangential estimates proved in arXiv:2103.10046.
Item Type:  Article 

Additional Information:  This paper is part of the earlier submission arXiv:1901.08261v2 
Uncontrolled Keywords:  Uniformly elliptic operators; elliptic measure; the Green function; 1sided nontangentially accessible domains; 1sided chordarc domains, capacity density condition; Ahlforsregularity; Muckenhoupt weights; Carleson measures; square function estimates; nontangential maximal function; dyadic analysis; sawtooth domains; perturbation 
Divisions:  Faculty of Science and Health Faculty of Science and Health > Mathematical Sciences, Department of 
SWORD Depositor:  Elements 
Depositing User:  Elements 
Date Deposited:  12 Sep 2019 15:36 
Last Modified:  07 Jul 2022 12:02 
URI:  http://repository.essex.ac.uk/id/eprint/25314 
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