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Harnack Inequality for a class of Degenerate Elliptic Equations in Non-Divergence Form

Abedin, Farhan
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Thesis/Dissertation
Date
2018
Advisor
Gutiérrez, Cristian E., 1950-
Committee member
Berhanu, Shiferaw
Yang, Wei-shih, 1954-
Futer, David
Hynd, Ryan
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Department
Mathematics
Subject
Mathematics
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http://hdl.handle.net/20.500.12613/624
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Abedin_temple_0225E_13387.pdf
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DOI
http://dx.doi.org/10.34944/dspace/606
Abstract
We provide two proofs of an invariant Harnack inequality in small balls for a class of second order elliptic operators in non-divergence form, structured on Heisenberg vector fields. We assume that the coefficient matrix is uniformly positive definite, continuous, and symplectic. The first proof emulates a method of E. M. Landis, and is based on the so-called growth lemma, which establishes a quantitative decay of oscillation for subsolutions. The second proof consists in establishing a critical density property for non-negative supersolutions, and then invoking the axiomatic approach developed by Di Fazio, Gutiérrez and Lanconelli to obtain Harnack’s inequality.
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