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Summary: MAXIMUM PRINCIPLES FOR A CLASS OF NONLINEAR
SECOND ORDER ELLIPTIC DIFFERENTIAL EQUATIONS
G. Porru, A. Tewodros and S. Vernier-Piro
Abstract. In this paper we investigate maximum principles for functionals de ned
on solutions to special partial di erential 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 = ux of the quasilinear second order equation
1:1
,
gq2ui
i = hq2
in domains RN. Here and in the sequel the subindex i i = 1;:::;N denotes
partial di erentiation with respect to xi, the summation convention from 1 to N
over repeated indices is in e ect, q2 = uiui, g and h are two smooth functions. In
order for equation 1.1 to be elliptic we suppose g 0 and G 0; where
1:2 G = g + 2g0:
Following Payne and Philippin 5,6,7 we derive some maximum principles for func-
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