First Eigenvalue Problem Of Dirichlet Forms

In the first two chapters, we study the first eigenfunction on a bounded domain ? with respect to a regular, strongly local Dirichlet form(E,F). Given a Poincaré inequality and volume doubling condition, using the idea of Mosco, we prove a localised Nash inequality.If reverse doubling condition also holds, we prove further Nash inequality. We obtain an L∞-estimate on bounded first eigenfunctions from Nash inequality.We prove a Harnack inequality for first eigenfunctions from a variant of Faber-Krahn inequality, a cutoff Sobolev inequality in annuli, doubling condition and reverse doubling condition.In the third chapter, we give the definition of a quasi-minima of Dirichlet forms, and prove its stability under the rough isometry of Dirichlet forms. We use a weak version of Faber-Krahn inequality and the cutoff Sobolev inequality in annuli to prove that any quasi-minma satisfies an L2-mean value inequality.