Doctor of Philosophy / Department of Mathematics / Ivan Blank / In the Fermi Lectures on the obstacle problem in 1998, Caffarelli gave a proof of the mean value theorem which extends to general divergence form uniformly elliptic operators. In the general setting, the result shows that for any such operator L and at any point [chi]₀ in the domain, there exists a nested family of sets { D[subscript]r([chi]₀) } where the average over any of those sets is related to the value of the function at [chi]₀. Although it is known that the { D[subscript]r([chi]₀) } are nested and are comparable to balls in the sense that there exists c, C depending only on L such that B[subscript]cr([chi]₀) ⊂ D[subscript]r([chi]₀) ⊂ B[subscript]Cr([chi]₀) for all r > 0 and [chi]₀ in the domain, otherwise their geometric and topological properties are largely unknown. In this work we begin the study of these topics and we prove a few results about the geometry of these sets and give a couple of applications of the theorems.
Identifer | oai:union.ndltd.org:KSU/oai:krex.k-state.edu:2097/36205 |
Date | January 1900 |
Creators | Aryal, Ashok |
Publisher | Kansas State University |
Source Sets | K-State Research Exchange |
Language | en_US |
Detected Language | English |
Type | Dissertation |
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