Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operators

• Published in: Transactions of the American Mathematical Society 2019-02, Vol.371 (4), p.2797-2835
• Main Authors: CAVERO, JUAN, HOFMANN, STEVE, MARTELL, JOSÉ MARÍA
• Format: Article
• Language: English
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/tran/7536

Let \Omega \subset \mathbb{R}^{n+1}, n\ge 2, be a 1-sided chord-arc domain; that is, a domain which satisfies interior corkscrew and Harnack chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path connectedness), and whose boundary \partial \Omega is n-dimensional Ahlfors regular. Consider L_0 and L two real symmetric divergence form elliptic operators, and let \omega _{L_0}, \omega _L be the associated elliptic measures. We show that if \omega _{L_0}\in A_\infty (\sigma ), where \sigma =H^n{\left \vert _{\,{\partial \Omega }}\right .}, and L is a perturbation of L_0 (in the sense that the discrepancy between L_0 and L satisfies certain Carleson measure condition), then \omega _L\in A_\infty (\sigma ). Moreover, if L is a sufficiently small perturbation of L_0, then one can preserve the reverse Hölder classes; that is, if for some 1