Heat trace asymptotics for domains with singular boundaries

Abstract

This dissertation consists of three main results regarding heat trace asymptotics for bounded domains with cusps. First, a refined asymptotic expansion for the Dirichlet heat trace θ(t) = Treᵗ(Δ)ᴰ on a planar domain with a single cusp is presented. First three terms appeared earlier in the physics literature. We prove them, together with logarithmic remainder estimate. Second, we obtain similar results for a family of three-dimensional solids of revolution with a cusp. Third, we calculate bounds for the Neumann heat trace Ψ(t) = Treᵗ(Δ)ᴺ on a planar region a cusp, which then allow us to conclude the first two terms in the asymptotic expansion of Ψ(t). For the upper bound, we use Golden-Thompson inequality. All the results for the Dirichlet heat trace and the calculation of the lower bound for the Neumann heat trace use Brownian motion.