Doctor of Philosophy / Department of Mathematics / Diego Maldonado / Originally introduced in 1961 by Carl Gustav Axel Harnack [36] in the context of harmonic
functions in R[superscript]2, the so-called Harnack inequality has since been established for solutions to a wide variety of different partial differential equations (PDEs) by mathematicians
at different times of its historical development. Among them, Moser's iterative scheme [47-49] and Krylov-Safonov's probabilistic method [43, 44] stand out as pioneering theories, both in terms of their originality and their impact on the study of regularity of solutions to PDEs.
Caffarelli's work [12] in 1989 greatly simplified Krylov-Safonov's theory and established Harnack's
inequality in the context of fully non-linear elliptic PDEs. In this scenario, Caffarelli
and Gutierrez's study of the linearized Monge-Ampere equation [15, 16] in 2002-2003 served
as a motivation for axiomatizations of Krylov-Safonov-Caffarelli theory [3, 25, 57]. The
main work in this dissertation is a new axiomatization of Krylov-Safonov-Caffarelli theory.
Our axiomatic approach to Harnack's inequality in spaces of homogeneous type has some distinctive features. It sheds more light onto the role of the so-called critical density property, a property which is at the heart of the techniques developed by Krylov and Safonov. Our structural assumptions become more natural, and thus, our theory better suited, in the context of variational PDEs. We base our method on the theory of Muckenhoupt's A[subscript]p weights. The dissertation also gives an application of our axiomatic approach to Harnack's inequality in the context of infinite graphs. We provide an alternate proof of Harnack's inequality for harmonic functions on graphs originally proved in [21].
| Identifer | oai:union.ndltd.org:KSU/oai:krex.k-state.edu:2097/14189 |
| Date | January 1900 |
| Creators | Silwal, Sharad Deep |
| Publisher | Kansas State University |
| Source Sets | K-State Research Exchange |
| Language | en_US |
| Detected Language | English |
| Type | Dissertation |
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