DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

MAXIMUM PRINCIPLES FOR ELLIPTIC EQUATIONS
IN UNBOUNDED DOMAINS
ANTONIO VITOLO
We investigate geometric conditions to have the maximum principle for linear secondorder
elliptic equations in unbounded domains. Next we show structure conditions for
nonlinear operators to get the maximum principle in the same domains, then we consider
viscosity solutions, for which we can establish at once a comparison principle when one
of the solutions is regular enough. We also note that the methods underlying the present
results can be used to obtain related Phragmén-Lindelöf principles.
Copyright © 2006 Antonio Vitolo. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let us consider the linear second-order elliptic operator
Lw = a ij (x)D ij w + b i (x)Dw + c(x)w, (1.1)
acting on a function space of twice differentiable functions D(Ω) ⊂ C 2 (Ω), where Ω is a
domain (open connected set) of R n . Here the matrix of principal coefficients a ij (x)willbe
taken definite positive and bounded (uniformly with respect to x), satisfying the uniform
ellipticity condition
λ|X| 2 ≤ a ij (x)X i X j ≤ Λ|X| 2 , X ∈ R n , (1.2)
with ellipticity constants 0