 
Summary: MAXIMUM PRINCIPLES FOR A CLASS OF NONLINEAR
SECOND ORDER ELLIPTIC DIFFERENTIAL EQUATIONS
G. Porru, A. Tewodros and S. VernierPiro
Abstract. In this paper we investigate maximum principles for functionals defined
on solutions to special partial differential equations of elliptic type, extending results
by Payne and Philippin. We apply such maximum principles to investigate one
overdetermined problem.
1. Introduction.
We consider classical solutions u = u(x) of the quasilinear second order equation
(1:1)
\Gamma
g(q 2 )u i
\Delta
i
= h(q 2 )
in
domains\Omega ae R N . Here and in the sequel the subindex i (i = 1; :::; N) denotes
partial differentiation with respect to x i , the summation convention (from 1 to N)
over repeated indices is in effect, q 2 = u i u i , g and h are two smooth functions. In
order for equation (1.1) to be elliptic we suppose g ? 0 and G ? 0; where
