Prime End Boundaries of Domains in Metric Spaces, and the Dirichlet Problem

Prime End Boundaries of Domains in Metric Spaces, and the Dirichlet Problem
By:Dewey Estep
Published on 2015 by

Let O be a domain in a metric measure space X of bounded geometry. In this thesis we define and investigate the prime end boundary bounded O, denoted PO, and attempt to solve the Dirichlet problem on said domains. We show that, in bounded O satisfying a certain key assumption, we may solve the Dirichlet problem with prime end boundary data f by using the Perron method and that such a solution coincides with the solution Hf given by the obstacle problem on O with obstacle -8. Here, our key assumption is that every end of O has a prime end of O which divides it. It is currently unknown if any bounded domains fails to satisfy this assumption. We also create a definition of prime ends for unbounded O. By using the sphericalization results of Li and Shanmugalingam in [20], we are able to show that the prime end boundary of an unbounded O is homeomorphic to the prime end boundary of the image of O under the sphericalization of X. We then show that we may solve the Dirichlet problem for such domains with prime end boundary data f by using the Perron method and that such a solution coincides with the solution Hf given by the appropriate obstacle problem, with the additional assumption that f-Hf extends p-quasicontinuously to 0 on PO.

This Book was ranked at 31 by Google Books for keyword prime.

Book ID of Prime End Boundaries of Domains in Metric Spaces, and the Dirichlet Problem's Books is UQM1tAEACAAJ, Book which was written byDewey Estephave ETAG "V7g0kuZ/Ngs"

Book which was published by since 2015 have ISBNs, ISBN 13 Code is and ISBN 10 Code is

Reading Mode in Text Status is false and Reading Mode in Image Status is false

Book which have "96 Pages" is Printed at BOOK under Category

This Book was rated by Raters and have average rate at ""

This eBook Maturity (Adult Book) status is NOT_MATURE

Book was written in en

eBook Version Availability Status at PDF is falseand in ePub is false

Book Preview