Document Type

Dissertation

Date of Degree

Fall 2012

Degree Name

PhD (Doctor of Philosophy)

Degree In

Mathematics

First Advisor

Lihe Wang

Abstract

In this study we provide a new proof of C¹,α boundary regularity for finite perimeter sets with flat boundary which are local minimizers of a variational mean curvature formula. Our proof is provided for curvature term HLΩ. The proof is a generalization of Cafarelli and C#243;rdoba's method, and combines techniques from geometric measure theory and the theory of viscosity solutions which have been developed in the last 50 years. We rely on the delicate interplay between the global nature of sets which are variational minimizers of a given functional, and the pointwise local nature of comparison surfaces which satisfy certain PDE. As a heuristic, in our proof we can consider the curvature as an error term which is estimated and controlled at each point of the calculation.

Keywords

Finite Perimeter, mean curvature, minimal surface, Regularity, Viscosity Solution

Pages

vi, 68 pages

Bibliography

Includes bibliographical references (pages 66-68).

Copyright

Copyright 2012 Stephen William Welch

Included in

Mathematics Commons

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