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Comparison of BV Norms in Weighted Euclidean Spaces and Metric Measure Spaces
CAMFIELD, CHRISTOPHER SCOTT

2008, PhD, University of Cincinnati, Arts and Sciences : Mathematical Sciences.
The study of functions of bounded variation has many applications, including the study of minimal surfaces, discontinuity hypersurfaces, nonlinear diffusion equations, and image segmentation. It is desirable to have a similar tool in spaces other than Euclidean domains. In recent years, some generalizations have been explored. In this thesis, we will examine a few proposed generalizations of the theory of functions of bounded variation. One, by Baldi, to weighted Euclidean spaces and another, by Miranda, to metric measure spaces are of particular interest. Since both are generalizations of the classical definition, they certainly agree with each other in Euclidean domains. They can both also be applied to weighted Euclidean spaces, so it is natural to ask whether these definitions in this setting are equivalent or even comparable. In all, four different norms are studied. When the weight is bounded above and bounded below away from zero, all four norms are comparable with the classical BV norm. We will give conditions that ensure the two definitions are equivalent. Examples are also constructed showing that the two norms are sometimes not even comparable. We will look at measure properties, give criteria for determining if a set is of weighted finite perimeter, and use the BV norms to construct vector-valued measures that provide a type of integration by parts formula. We also study Miranda's definition in the setting of a metric space satisfying a doubling condition and a Poincare Inequality. Here we show that the BV definition in this setting is a relaxation of the norm of the (1,1)-Newtonian space. We also explore the idea of using finite dimensional D-structures in metric spaces developed by Cheeger to construct a BV norm.
Nageswari Shanmugalingam, PhD (Committee Chair)
David Herron, PhD (Committee Member)
David Minda, PhD (Committee Member)
Michele Miranda, PhD (Committee Member)
Xiao Zhong, PhD (Committee Member)
141 p.

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CAMFIELD, C. (2008). Comparison of BV Norms in Weighted Euclidean Spaces and Metric Measure Spaces. (Electronic Thesis or Dissertation). Retrieved from https://etd.ohiolink.edu/

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CAMFIELD, CHRISTOPHER. "Comparison of BV Norms in Weighted Euclidean Spaces and Metric Measure Spaces." Electronic Thesis or Dissertation. University of Cincinnati, 2008. OhioLINK Electronic Theses and Dissertations Center. 24 Nov 2017.

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CAMFIELD, CHRISTOPHER "Comparison of BV Norms in Weighted Euclidean Spaces and Metric Measure Spaces." Electronic Thesis or Dissertation. University of Cincinnati, 2008. https://etd.ohiolink.edu/

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