Comparison of BV Norms in Weighted Euclidean Spaces and Metric Measure Spaces
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University of Cincinnati / OhioLINK
2008
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ndltd-OhioLink-oai-etd.ohiolink.edu-ucin12115515792021-08-03T06:12:38Z Comparison of BV Norms in Weighted Euclidean Spaces and Metric Measure Spaces CAMFIELD, CHRISTOPHER SCOTT Mathematics functions of bounded variation metric measure spaces weighted Euclidean spaces 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. 2008-08-25 English text University of Cincinnati / OhioLINK http://rave.ohiolink.edu/etdc/view?acc_num=ucin1211551579 http://rave.ohiolink.edu/etdc/view?acc_num=ucin1211551579 unrestricted This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws. |
| collection |
NDLTD |
| language |
English |
| sources |
NDLTD |
| topic |
Mathematics functions of bounded variation metric measure spaces weighted Euclidean spaces |
| spellingShingle |
Mathematics functions of bounded variation metric measure spaces weighted Euclidean spaces CAMFIELD, CHRISTOPHER SCOTT Comparison of BV Norms in Weighted Euclidean Spaces and Metric Measure Spaces |
| author |
CAMFIELD, CHRISTOPHER SCOTT |
| author_facet |
CAMFIELD, CHRISTOPHER SCOTT |
| author_sort |
CAMFIELD, CHRISTOPHER SCOTT |
| title |
Comparison of BV Norms in Weighted Euclidean Spaces and Metric Measure Spaces |
| title_short |
Comparison of BV Norms in Weighted Euclidean Spaces and Metric Measure Spaces |
| title_full |
Comparison of BV Norms in Weighted Euclidean Spaces and Metric Measure Spaces |
| title_fullStr |
Comparison of BV Norms in Weighted Euclidean Spaces and Metric Measure Spaces |
| title_full_unstemmed |
Comparison of BV Norms in Weighted Euclidean Spaces and Metric Measure Spaces |
| title_sort |
comparison of bv norms in weighted euclidean spaces and metric measure spaces |
| publisher |
University of Cincinnati / OhioLINK |
| publishDate |
2008 |
| url |
http://rave.ohiolink.edu/etdc/view?acc_num=ucin1211551579 |
| work_keys_str_mv |
AT camfieldchristopherscott comparisonofbvnormsinweightedeuclideanspacesandmetricmeasurespaces |
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1719432759414358016 |