2026-09-23T08:45:36Zhttps://api.repository.cam.ac.uk/server/oai/requestoai:www.repository.cam.ac.uk:1810/2739392024-06-26T13:53:47Zcom_1810_246436com_1810_256064col_1810_246437
Boundary Value Problems for the Laplace Equation on Convex Domains with Analytic Boundary
10.17863/CAM.21012
Rockstroh , Parousia
Ashton , Anthony
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.
University of Cambridge
2018-04-07
Thesis
Doctoral
Doctor of Philosophy (PhD)
en
Cambridge Trust
https://www.repository.cam.ac.uk/handle/1810/273939
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Partial Differential Equations
Laplace Equation
Fokas Method
Unified Transform Method
Elliptic PDEs
Numerical Method