First Eigenvalue of the Dirichlet-Laplacian and its Variations

In this talk, we'll discuss eigenvalues and eigenfunctions of the laplacian on open sets, $\Omega \subseteq \R^n$, subject to the dirichlet condition on $\partial \Omega$. In particular, we'll prove the Faber-Krahn theorem, which states that the minimizer of $\lambda_1(\Omega)$ for fixed volume $|\Omega| = c$ is achieved by the euclidean ball of volume, $c$. If time permits, we'll compute variations of \lambda_1, with respect to domain variations $\{\Omega_t\}$