Structure of eigenvalues and eigenfunctions

傅里叶展开是偏微分方程分析中的一个重要工具。不规则函数仍然可以展开为傅立叶级数，然后通过截断傅立叶级数，得到收敛于原始不规则函数的光滑函数序列。这个想法-以及相关的技术-形成了该领域许多证明的关键部分。然而，截断傅立叶级数-更一般地，截断本征函数展开-可能不会收敛于Lp，p不等于2，这在许多情况下可能会证明是有问题的。基于观察到当p不为2时，精心选择的傅立叶级数截断确实收敛于Lp，本项目将研究在一般域中的本征函数展开式上可以携带多少（傅立叶级数的情况对应于平方域中的拉普拉斯算子）。我们已经获得了新的收敛结果，某些类型的三角域，并正在开发一种技术，用于分析收敛，因为我们变形的域控制的方式。该研究属于数学分析领域，在连续介质力学、流体动力学和空气动力学的更多理论方面具有潜在的应用。

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.