Electronic Journal of Differential Equations,
Vol. 1999(1999), No. 38, pp. 1-20.
Title: Boundary behavior and estimates for solutions
of equations containing the $p$-laplacian
Authors: Jacqueline Fleckinger (Univ. Toulouse-1, Toulouse, France)
Evans M. Harrell II (Georgia Tech, Atlanta, GA, USA)
Francois de Thelin (Univ. Paul Sabatier, Toulouse, France)
Abstract:
We use ``Hardy-type'' inequalities to derive $L^q$ 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 $L^q$ 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 $L^q$ 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.