Capacitary Estimates for Dirichlet Eigenvalues

✍ Scribed by I. McGillivray

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 591 KB
• Volume: 139
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.0085

✦ Synopsis

Let H be the non-negative definite selfadjoint operator associated to a regular irreducible Dirichlet form on L 2 (X, m). Assume that H has discrete spectrum. We study perturbations of this operator which arise through the imposition of Dirichlet boundary conditions on a compact subset of X. The eigenvalues of the perturbed operator are of course shifted to the right. Under an ultracontractivity condition, we show that the magnitude of this shift can be estimated by the capacity. We also obtain a capacitary lower bound for the ground-state shift under suitable conditions. An application to the ``crushed ice'' problem is described. 1996 Academic Press, Inc. For arbitrary compact K, estimates in terms of the capacity are given in [3].

In the following Section we present essential tools from the theory of probabilistic potential theory. Sections 3 and 4 are devoted to the proof of the article no.