## Differential and Integral Equations

### A priori estimate for the first eigenvalue of the $p$-Laplacian

Ryuji Kajikiya

#### Abstract

We study the first eigenvalue of the $p$-Laplacian under the Dirichlet boundary condition. For a convex domain, we give an a priori estimate for the first eigenvalue in terms of the radius $d$ of the maximum ball contained in the domain. As a consequence, we prove that the first eigenvalue diverges to infinity as $p\to\infty$ if the domain is convex and $d\leq 1$. Moreover, we show that in the annulus domain $a < |x| < b$, the first eigenvalue diverges to infinity if $b-a\leq 2$ and converges to zero if $b-a>2$.

#### Article information

Source
Differential Integral Equations, Volume 28, Number 9/10 (2015), 1011-1028.

Dates
First available in Project Euclid: 23 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.die/1435064548

Mathematical Reviews number (MathSciNet)
MR3360728

Zentralblatt MATH identifier
1363.35096

#### Citation

Kajikiya, Ryuji. A priori estimate for the first eigenvalue of the $p$-Laplacian. Differential Integral Equations 28 (2015), no. 9/10, 1011--1028. https://projecteuclid.org/euclid.die/1435064548