Short Talk: Brian Chao

The principal Dirichlet Laplacian eigenfunction $\varphi_U$ describes the invariant measure of a Brownian motion conditioned to remain inside a bounded domain $U\subseteq \mathbb{R}^n$. In this talk, I will present results about how domain perturbations affect $\varphi_U$. These results lead to explicit expressions for $\varphi_U$ in certain Euclidean domains, thus implying improved Dirichlet heat kernel estimates. This is joint work with Laurent Saloff-Coste.