Sharp Estimates for Dirichlet Eigenfunctions in Simply Connected Domains

✍ Scribed by R. Bañuelos

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 580 KB
• Volume: 125
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1996.0032

✦ Synopsis

Introduction

The following result on the exponential decay of the eigenfunctions for the Dirichlet Laplacian in horn shaped regions is proved in R. Ban~uelos and B. Davis [5], [6].

Theorem A. Let %: (0, ) Ä (0, 1] be continuous and define

Suppose %(x) a 0 as x A and let . * be any L 2 -eigenfunction for the Dirichlet Laplacian in D % with eigenvalue *. Then for any =>0, | D% |. * (x, y)| 2 e (4&=)?Â4 0 x :*(s) ds dx dy< , (0.1)

If in addition % # L 1 (dx) then we may take ==0. However, in general = cannot be taken to be 0.

The purpose of this paper is to prove a version of Theorem A for more general simply connected domains in the plane. Before we state this precisely we introduce some notation. If 0 is a simply connected domain in the plane and z # 0, we define the density of the hyperbolic metric at z by _ 0 (z)=1Â|F$(0)| where F is a conformal mapping sending the unit disc, henceforth denoted by D, onto 0 with F(0)=z. Equivalently, _ 0 (z)= |G$(z)|