Integrals of harmonic functions over curves and surfaces
Movshovich, Yevgenya E.
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https://hdl.handle.net/2142/19848
Description
Title
Integrals of harmonic functions over curves and surfaces
Author(s)
Movshovich, Yevgenya E.
Issue Date
1995
Doctoral Committee Chair(s)
Miles, Joseph B.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
This thesis explores a new approach, begun by Maurice Heins and Jang-Mei Wu, to studying the near-boundary behavior of positive solutions of elliptic DE's by finding surfaces which minimize the integrals of solutions over the family of all closed surfaces or curves tending to the boundary. The inferior limit of integrals over this family is estimated for harmonic functions given by the Cantors measures on the unit circle in terms of the porosity of the Cantor set. The exact lower bound of it for all normalized positive harmonic functions in the unit ball of $R\sp{n}$ is established. Also, its value for the functions given by the Dirac measure on the unit n-sphere is computed. The generalized formula of the surface area element in spherical coordinates for the dimension $n>3$ is derived.
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