Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Authors: Juan Cavero, Steve Hofmann and José María Martell
Journal: Trans. Amer. Math. Soc. 371 (2019), 2797-2835
MSC (2010): Primary 31B05, 35J08, 35J25; Secondary 42B99, 42B25, 42B37
DOI: https://doi.org/10.1090/tran/7536
Published electronically: October 2, 2018
MathSciNet review: 3896098
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: 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<p<\infty $, one has $ \omega _{L_0}\in RH_p(\sigma )$, then $ \omega _{L}\in RH_p(\sigma )$. Equivalently, if the Dirichlet problem with data in $ L^{p'}(\sigma )$ is solvable for $ L_0$, then it is for $ L$ also. These results can be seen as extensions of the perturbation theorems obtained by Dahlberg; Fefferman, Kenig, and Pipher; and Milakis, Pipher, and Toro in more benign settings. As a consequence of our methods, we can show that for any perturbation of the Laplacian (or, more in general, of any elliptic symmetric operator with Lipschitz coefficients satisfying certain Carleson condition) if its elliptic measure belongs to $ A_\infty (\sigma )$, then necessarily $ \Omega $ is in fact a nontangentially accessible domain (and hence chord-arc), and therefore its boundary is uniformly rectifiable.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 31B05, 35J08, 35J25, 42B99, 42B25, 42B37

Retrieve articles in all journals with MSC (2010): 31B05, 35J08, 35J25, 42B99, 42B25, 42B37


Additional Information

Juan Cavero
Affiliation: Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, E-28049 Madrid, Spain
Email: juan.cavero@icmat.es

Steve Hofmann
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: hofmanns@missouri.edu

José María Martell
Affiliation: Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, 13-15, E-28049 Madrid, Spain
Email: chema.martell@icmat.es

DOI: https://doi.org/10.1090/tran/7536
Keywords: Elliptic measure, Poisson kernel, Carleson measures, $A_\infty$ Muckenhoupt weights
Received by editor(s): August 22, 2017
Received by editor(s) in revised form: January 9, 2018
Published electronically: October 2, 2018
Additional Notes: The first author was partially supported by la Caixa–Severo Ochoa international PhD Programme. The first and third authors acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2015-0554). They also acknowledge that the research leading to these results received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT
The second author was supported by NSF grant DMS-1664047.
Article copyright: © Copyright 2018 American Mathematical Society