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

Estep, Dewey
http://orcid.org/0000-0001-5740-1540
http://rave.ohiolink.edu/etdc/view?acc_num=ucin1439295199

Abstract Details

2015, PhD, University of Cincinnati, Arts and Sciences: Mathematical Sciences.
Let Ω be a domain in a metric measure space X of bounded geometry. In this thesis we define and investigate the prime end boundary bounded Ω, denoted ∂PΩ, and attempt to solve the Dirichlet problem on said domains. We show that, in bounded Ω 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 Ω with obstacle -∞. Here, our key assumption is that every end of Ω has a prime end of Ω 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 Ω. By using the sphericalization results of Li and Shanmugalingam in [20], we are able to show that the prime end boundary of an unbounded Ω is homeomorphic to the prime end boundary of the image of Ω 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 ∂PΩ.
Nageswari Shanmugalingam, Ph.D. (Committee Chair)
Thomas James Bieske, Ph.D. (Committee Member)
Michael Goldberg, Ph.D. (Committee Member)
David Herron, Ph.D. (Committee Member)
Carl David Minda, Ph.D. (Committee Member)
95 p.
Mathematics
Perron method; Prime End; Metric Measure Space; DIrichlet Problem

Recommended Citations

Citations

  • Estep, D. (2015). Prime End Boundaries of Domains in Metric Spaces and the Dirichlet Problem [Doctoral dissertation, University of Cincinnati]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1439295199

    APA Style (7th edition)

  • Estep, Dewey. Prime End Boundaries of Domains in Metric Spaces and the Dirichlet Problem. 2015. University of Cincinnati, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ucin1439295199.

    MLA Style (8th edition)

  • Estep, Dewey. "Prime End Boundaries of Domains in Metric Spaces and the Dirichlet Problem." Doctoral dissertation, University of Cincinnati, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1439295199

    Chicago Manual of Style (17th edition)

ucin1439295199
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This open access ETD is published by University of Cincinnati and OhioLINK.