Boundary Value Problems for the Laplace Equation on Convex Domains with Analytic Boundary
In this thesis we study boundary value problems for the Laplace equation on do mains with smooth boundary. Central to our analysis is a relation, known as the global relation, that couples the boundary data for a given BVP. Previously, the global re lation has primarily been applied to elliptic PDEs deﬁned on polygonal domains. In this thesis we extend the use of the global relation to domains with smooth boundary. This is done by introducing a new transform, denoted by F_p, that is an analogue of the Fourier transform on smooth convex curves. We show that the F_p-transform is a bounded and invertible integral operator. Following this, we show that the F_p-transform naturally arises in the global relation for the Laplace equation on domains with smooth boundary. Using properties of the F_p-transform, we show that the global relation deﬁnes a continuously invertible map between the Dirichlet and Neumann data for a given BVP for the Laplace equation. Following this, we construct a numerical method that uses the global relation to ﬁnd the Neumann data, given the Dirichlet data, for a given BVP for the Laplace equation on a domain with smooth boundary.
Partial Differential Equations, Laplace Equation, Fokas Method, Unified Transform Method, Elliptic PDEs, Numerical Method
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