Structure of eigenvalues and eigenfunctions

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Fourier expansions are a key tool in the analysis of partial differential equations. Irregular functions can still be expanded as Fourier series, and then by truncating the Fourier series one obtains a sequence of smooth functions that converges to the original irregular function. This idea - and related techniques - forms a key part of many proofs in the area. However, truncated Fourier series - and, more generally, truncated eigenfunction expansions - may not converge in Lp for p not equal to 2, and this can prove problematic in many situations. Based on the observation that carefully chosen truncations of Fourier series do converge in Lp when p is not 2, this project will investigate how much this can be carried over the eigenfunction expansions in general domains (the Fourier series case corresponding to the Laplacian in a square domain). We have obtained new convergence results for certain types of triangular domains and are developing a technique for analysing convergence as we deform the domain in controlled ways. The research lies in the area of Mathematical Analysis, with potential applications in the more theoretical aspects of Continuum Mechanics and Fluid dynamics and aerodynamics.