In this project we will attack a long-standing conjecture in potential theory, due to Carlos Kenig in 1994, regarding the so-called "double-layer potential operator" K on a closed curve C in the plane. (Potential theory is the study of solutions to Laplace's equation and is a fundamental topic in PDEs and in analysis more generally. The double-layer potential operator is a key singular integral operator that arises in this theory, for example when formulating the Dirchlet problem in a bounded domain as a boundary integral equation) Kenig's conjecture is that the so-called essential spectral radius of K is < 1/2 as an operator on the space of functions that are square integrable on C with respect to surface measure. The spectrum of a (bounded linear) operator K is the set of complex values z for which zI-K is not invertible, where I denotes the identity operator. The essential spectrum is arguably the most important part of the spectrum for a bounded operator on a Hilbert space, it is the set of z for which zI-K is not even approximately invertible in a sense that can be made precise. Kenig's conjecture has been shown to hold when C is the boundary of a convex domain and when C is a polygon. In this project we will explore whether the conjecture holds for much more complicated boundaries. To do this we will use a combination of methods, which will depend on the background and interests of the candidate but will include analysis and functional analysis methods and rigorously justified numerical analysis and computation of the essential spectrum for specific examples in Matlab or Python.
The candidate will work with and learn from the supervisors and will also have opportunities to interact with and visit other leading experts in this area, for example Euan Spence (Bath) and Karl-Mikael Perfekt (Trondheim). They will also present their progress at international conferences.
A BSc (2:1 or above) or MSc in Mathematics, Statistics, Computer Science subjects. Some experience in programming with C/C++, Matlab, or any other programming languages is preferred.
Informal enquiries can be made to: Simon Chandler-Wilde at [Email Address Removed]
More detail about the project is available through the web link below:
https://research.reading.ac.uk/pure-and-applied-analysis/phd-projects-in-analysis/
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