Asymptotic results concerning heat content and spectra of the Laplacian

We investigate the relationship between analytical quantities associated with the Laplacian on a domain and the geometry of . In particular, we prove new results concerning small-time asymptotics for the heat content of polygons contained inside larger polygons with Neumann boundary conditions imposed. We also prove some new results concerning the asymptotic behaviour of minimisers to spectral shape optimisation problems for Neumann, and consequently Robin, eigenvalues of the Laplacian under perimeter and diameter constraint. Moreover, we consider some related spectral shape optimisation problems for mixed Dirichlet-Neumann, so-called Zaremba, eigenvalues of the Laplacian.