Perturbations of elliptic operators in 1-sided chord-arc domains. Part II: Non-symmetric operators and Carleson measure estimates

Autor: José María Martell, Juan Cavero, Tatiana Toro, Steve Hofmann
Přispěvatelé: Ministerio de Economía y Competitividad (España), European Commission
Rok vydání: 2020
Předmět:
Zdroj: Digital.CSIC. Repositorio Institucional del CSIC
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ISSN: 1088-6850
0002-9947
DOI: 10.1090/tran/8148
Popis: We generalize to the setting of 1-sided chord-arc domains, that is, to domains satisfying the interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness) and which have an Ahlfors regular boundary, a result of Kenig-Kirchheim-Pipher-Toro, in which Carleson measure estimates for bounded solutions of the equation $ Lu=-\operatorname {div}(A\nabla u) = 0$ with $ A$ being a real (not necessarily symmetric) uniformly elliptic matrix imply that the corresponding elliptic measure belongs to the Muckenhoupt $ A_\infty $ class with respect to surface measure on the boundary. We present two applications of this result. In the first one we extend a perturbation result recently proved by Cavero-Hofmann-Martell presenting a simpler proof and allowing non-symmetric coefficients. Second, we prove that if an operator $ L$ as above has locally Lipschitz coefficients satisfying certain Carleson measure condition, then $ \omega _L\in A_\infty $ if and only if $ \omega _{L^\top }\in A_\infty $. As a consequence, we can remove one of the main assumptions in the non-symmetric case of a result of Hofmann-Martell-Toro and show that if the coefficients satisfy a slightly stronger Carleson measure condition the membership of the elliptic measure associated with $ L$ to the class $ A_\infty $ yields that the domain is indeed a chord-arc domain.
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. The fourth author was partially supported by the Craig McKibben & Sarah Merner Professor in Mathematics, by NSF grant number DMS-1664867, and by the Simons Foundation Fellowship 614610.
Databáze: OpenAIRE
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