Electron. J. Diff. Eqns., Vol. 1999(1999), No. 38, pp. 1-20.

Boundary behavior and estimates for solutions of equations containing the p-laplacian

Jacqueline Fleckinger, Evans M. Harrell II, & Francois de Thelin

Abstract:
We use ``Hardy-type'' inequalities to derive Lq estimates for solutions of equations containing the p-Laplacian with p>1. We begin by deriving some inequalities using elementary ideas from an early article [B3] which has been largely overlooked. Then we derive Lq estimates of the boundary behavior of test functions of finite energy, and consequently of principal (positive) eigenfunctions of functionals containing the p-Laplacian. The estimates contain exponents known to be sharp when p=2. These lead to estimates of the effect of boundary perturbation on the fundamental eigenvalue. Finally, we present global Lq estimates of solutions of the Cauchy problem for some initial-value problems containing the p-Laplacian.

Submitted July 13, 1999. Published September 28, 1999.
Math Subject Classifications: 35J60, 35J70.
Key Words: p-Laplacian, Hardy inequlity,principal eigenvalue, boundary estimate, boundary perturbation.

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An erratum was attached to this article on April 28, 2003. See the last page to this manuscript.

Jacqueline Fleckinger
CEREMATH & UMR MIP, Universite Toulouse-1
21 allees de Brienne
31000 Toulouse, France
e-mail address: jfleck@univ-tlse1.fr
Evans M. Harrell II
School of Mathematics, Georgia Tech
Atlanta, GA 30332-0160, USA, and
UMR MIP, Universite Paul Sabatier
31062 Toulouse, France
e-mail address: harrell@math.gatech.edu
http://www.math.gatech.edu/~harrell/
Francois de Thelin
UMR MIP, Universite Paul Sabatier
31062 Toulouse, France
e-mail address: dethelin@mip.ups-tlse.fr

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