CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Seja Ω um domÃnio fixado em Rn com fronteira S de classe C2 e denote Ω′ = Rn Ω. Ambos Ω e Ω′ nÃo necessariamente conexos. Nessas condiÃÃes, pretendemos resolver os problemas de Dirichlet e Neumann. No intuito da resoluÃÃo dos problemas citados, faremos um estudo daTeoria de Fredholm (operadores compactos), bem como da transformada de Kelvin, harmonicidade no infinito e dos potenciais de camada. / Let Ω be a fixed domain in Rn with boundary S of class C2 and denote Ω′ = Rn Ω. Both Ω and Ω′ not necessarily connected. Under these conditions, we intend to solve the problems of Dirichlet and Neumann. In order to overcome the mentioned the problems, we will study the Fredholm theory (compact operators), the Kelvin transformed, harmonicity in the infinite and potential of the layer.

Thesis. 1975. Ph.D.--Massachusetts Institute of Technology. Dept. of Meteorology. / Vita. / Bibliography: leaves 168-170. / Ph.D.

We give a method, by solving a nonlinear system of equations, for Gauss harmonic interpolation formulas which are useful for approximating, the solution of the Dirichlet problem. We also discuss approximations for integrals of the form I(f) = (1/2πi) ∫_L (f(z)/(z-α)) dz. Our approximations shall be of the form Q(f) = Σ_k=1ⁿ A_kf(τ_k). We characterize the nodes τ₁, τ₂, …, τ_n, to get the maximum precision for our formulas. Finally, we propose a general problem of approximating for linear functionals; our results need further development. / Ph. D.

This paper deals with approximating an upper bound for the number of equilibrium points of a potential field produced by point charges in the plane. This is a simplified form of a problem posed by Maxwell [4], who considered spatial configurations of the point charges. Using algebraic techniques, we will give an upper bound for planar charges that is sharper than the bound given in [6] for most general configurations of charges. Then we will study an example of a configuration of charges that has exactly the number of equilibrium points that Maxwell's conjecture predicts, and we will look into the nature of the extremal points in this case. We will conclude with a solution to the twin problem for the logarithmic potential, followed by a discussion of the conditions necessary for a degenerate case in the plane.

University of Macau / Faculty of Science and Technology / Department of Mathematics

In this thesis, we examined some Dirichlet type problems of the form: \begin{eqnarray*} \triangle u & = & 0\ {\rm in\ } \mathbb{R}^n \\ u & = & f\ {\rm on\ } \psi = 0, \end{eqnarray*} and we were particularly interested in finding entire solutions when entire data was prescribed. This is an extension of the work of D. Siegel, M. Mouratidis, and M. Chamberland, who were interested in finding polynomial solutions when polynomial data was prescribed. In the cases where they found that polynomial solutions always existed for any polynomial data, we tried to show that entire solutions always existed given any entire data. For half space problems we were successful, but when we compared this to the heat equation, we found that we needed to impose restrictions on the type of data allowed. For problems where data is prescribed on a pair of intersecting lines in the plane, we found a surprising dependence between the existence of an entire solution and the number theoretic properties of the angle between the lines. We were able to show that for numbers $\alpha$ with $\omega_1$ finite according to Mahler's classification of transcendental numbers, there will always be an entire solution given entire data for the angle $2\alpha\pi$ between the lines. We were also able to construct an uncountable, dense set of angles of measure 0, much in the spirit of Liouville's number, for which there will not always be an entire solution for all entire data. Finally, we investigated a problem where data is given on the boundary of an infinite strip in the plane. We were unable to settle this problem, but we were able to reduce it to other {\it a priori} more tractable problems.

In this thesis, we examined some Dirichlet type problems of the form: \begin{eqnarray*} \triangle u & = & 0\ {\rm in\ } \mathbb{R}^n \\ u & = & f\ {\rm on\ } \psi = 0, \end{eqnarray*} and we were particularly interested in finding entire solutions when entire data was prescribed. This is an extension of the work of D. Siegel, M. Mouratidis, and M. Chamberland, who were interested in finding polynomial solutions when polynomial data was prescribed. In the cases where they found that polynomial solutions always existed for any polynomial data, we tried to show that entire solutions always existed given any entire data. For half space problems we were successful, but when we compared this to the heat equation, we found that we needed to impose restrictions on the type of data allowed. For problems where data is prescribed on a pair of intersecting lines in the plane, we found a surprising dependence between the existence of an entire solution and the number theoretic properties of the angle between the lines. We were able to show that for numbers $\alpha$ with $\omega_1$ finite according to Mahler's classification of transcendental numbers, there will always be an entire solution given entire data for the angle $2\alpha\pi$ between the lines. We were also able to construct an uncountable, dense set of angles of measure 0, much in the spirit of Liouville's number, for which there will not always be an entire solution for all entire data. Finally, we investigated a problem where data is given on the boundary of an infinite strip in the plane. We were unable to settle this problem, but we were able to reduce it to other {\it a priori} more tractable problems.

Thesis (Ph. D.)--Brigham Young University. Dept. of Mathematics, 2004. / Includes bibliographical references (p. 113-117).

No description available.

Thesis (Ph. D.)--University of Washington, 2007. / Vita. Includes bibliographical references (p. 85-86).