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Sharp Estimates for Dirichlet Eigenfunctions in Horn ... - Project Euclid

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(^(x,0)>C(9(x)e 2 i ^(s) for x > 1. Our proof of the upper bound, which is very elementary, is based on the well- known method of Carleman which is widely used ...

C o m m u n i c a t i o n s IΠ

Commun. Math. Phys. 150, 209-215 (1992)

Mathematical Physics

© Springer-Verlag 1992

Sharp Estimates for Dirichlet Eigenfunctions in Horn-Shaped Regions Rodrigo Banuelos1 * and Burgess Davis 2 * 1 2

Department of Mathematics, Purdue University, W. Lafayette, IN 47907, USA Department of Statistics, Purdue University, W. Lafayette, IN 47907, USA

Received April 28, 1992

Abstract. We prove sharp exponential decay for the Dirichlet eigenfunctions in hornshaped regions in the plane. The estimate is obtained using a method of Carleman which is widely used in the study of harmonic measure. Such estimates can be applied to study the intrinsic ultracontractivity properties of the heat semigroup for such regions.

0. Introduction The purpose of this paper is to prove sharp exponential bounds for the decay of the Dirichlet eigenfunctions in "infinite trumpets" or "horn-shaped" regions. Such estimates, besides being of interest in their own right, can be used to prove intrinsic ultracontractivity, IU, for the Dirichlet Laplacian for these regions as in Davies and Simon [4]. Also, our result sharpens Theorem 7.3 in Davies and Simon [4]. Let #:(0, oo) —> [0,1] be continuous and let Dθ = {z = ( z , y):x>0,

-Θ(x) xx. Applying Green's theorem and using again the fact that φx vanishes on dDθ we find that xλ

(x)

x, y)]dx dy

/

-6 2

2

2max /" / |Vy»A(x,j/)| cixciy, / \ψx{x,y)\ dxdy\

= 2λ

and this completes the proof of the lemma. Proof of (a). To prove part (a) it is enough to estimate oo

/

θ(x) r

I

xX -θ(x)

x (1—ε)π J 2

\φx(x,y)\ e

x

*

αx(s)ds

(1.9) dxdy.

Shaφ Estimates for Dirichlet Eigenfunctions in Horn-Shaped Regions

213

Substituting oo

(χ) = Uχ

u'x(t)dt x

and applying Lemma5.1 we find that the expression in (1.9) is dominated by OO

/

/

/

x^

xx

OO

*

r

/e

\

—π J ax(s)ds Xχ

\χ

\ (

dx

dt I e /

x OO /

OO

t

\

dt \e

xχ

dx,

since ^(x) | 0, ax(s) | and we have that the last expression is

j e επ( ί E - a ; /

OO

f e-π

I— a (i

J X

2λ

x

X oo

ί

which is finite for any ε > 0 and we have proved the first assertion in (a). ι If θ G L (dx) and we take ε = 0 the above calculation shows that oo

θ(x)

oo

dxdy /

4λ π

/