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Problems In Classical Potential Theory With Applications To Mathematical Physics, Erik Lundberg
Problems In Classical Potential Theory With Applications To Mathematical Physics, Erik Lundberg
USF Tampa Graduate Theses and Dissertations
In this thesis we are interested in some problems regarding harmonic functions. The topics are divided into three chapters.
Chapter 2 concerns singularities developed by solutions of the Cauchy problem for a holomorphic elliptic equation, especially Laplace's equation. The principal motivation is to locate the singularities of the Schwarz potential. The results have direct applications to Laplacian growth (or the Hele-Shaw problem).
Chapter 3 concerns the Dirichlet problem when the boundary is an algebraic set and the data is a polynomial or a real-analytic function. We pursue some questions related to the Khavinson-Shapiro conjecture. A main topic of interest is …