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On Holder continuity of weak solutions to degenerate linear elliptic partial differential equations

For degenerate elliptic partial differential equations, it is often desirable to show that a weak solution is smooth. The first and most difficult step in this process is establishing local Hölder continuity. Sufficient conditions for establishing continuity have already been documented in [FP], [SW1], and [MRW], and their necessity in [R]. However, the complexity of the equations discussed in those works makes it difficult to understand the core structure of the arguments employed. Here, we present a harmonic-analytic method for establishing Hölder continuity of weak solutions in context of a simple linear equation
div(Q?u) = f
in a homogeneous space structure in order to showcase the form of the argument. Ad- ditionally, we correct an oversight in the adaptation of the John-Nirenberg inequality presented in [SW1], restricting it to a much smaller class of balls.

Identiferoai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:NSHD.ca#10222/35442
Date13 August 2013
CreatorsMombourquette, Ethan
Source SetsLibrary and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada
LanguageEnglish
Detected LanguageEnglish

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