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Boundary Harnack inequality for the linearized Monge-Ampère equations and applications


Author: Nam Q. Le
Journal: Trans. Amer. Math. Soc. 369 (2017), 6583-6611
MSC (2010): Primary 35B51, 35B65, 35J08, 35J96, 35J70
DOI: https://doi.org/10.1090/tran/7220
Published electronically: May 5, 2017
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Abstract: In this paper, we obtain boundary Harnack estimates and comparison theorem for nonnegative solutions to the linearized Monge-Ampère equations under natural assumptions on the domain, Monge-Ampère measures and boundary data. Our results are boundary versions of Caffarelli and Gutiérrez's interior Harnack inequality for the linearized Monge-Ampère equations. As an application, we obtain sharp upper bound and global $ L^p$-integrability for Green's function of the linearized Monge-Ampère operator.


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Nam Q. Le
Affiliation: Department of Mathematics, Indiana University, 831 E 3rd Street, Bloomington, Indiana 47405
Email: nqle@indiana.edu

DOI: https://doi.org/10.1090/tran/7220
Keywords: Boundary Harnack inequality, linearized Monge-Amp\`ere equation, Green's function, boundary localization theorem, Monge-Amp\`ere equation
Received by editor(s): June 3, 2016
Published electronically: May 5, 2017
Additional Notes: The research of the author was supported in part by the National Science Foundation under grant DMS-1500400.
Article copyright: © Copyright 2017 American Mathematical Society