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 H∈LΩ. 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