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    Examples of discontinuity for the variational solution of the minimal surface equation with Dirichlet data on a domain with a nonconvex corner and locally negative mean curvature

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    t13091_Melin.pdf (294.6Kb)
    Date
    2013-12
    Author
    Melin, Jaron Patric
    Advisor
    Lancaster, Kirk E.
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    Abstract
    The purpose of this thesis is to investigate the role of smoothness, specifically the smoothness of the boundary ∂Ω, in the behavior of the variational solution f on a domain Ω to the Dirichlet problem for the Minimal Surface Equation at a point O ∈ ∂Ω when the (generalized) curvature of ∂Ω has a negative upper bound in a neighborhood of O. We give examples which show that the assumption of boundary-regularity which Simon made in [12] or at least some weaker boundary-regularity assumption which excludes nonconvex corners in the boundary of the domain is necessary in order to guarantee that the variational solution of the Dirichlet problem for the Minimal Surface Equation is continuous in the closure of the domain for every Lipschitz-continuous boundary-data function ϕ : ∂Ω → R. This is independent of whether or not f equals ϕ on ∂Ω. Furthermore, these examples give credence to the Concus-Finn Conjecture, which still awaits to be proven in the case that the contact-angle is 0 or π at nonconvex corners.
    Description
    Thesis (M.S.)--Wichita State University, College of Liberal Arts and Sciences, Dept. of Mathematics, Statistics, and Physics
    URI
    http://hdl.handle.net/10057/10639
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