Co. Horgan et Le. Payne, PHRAGMEN-LINDELOF TYPE RESULTS FOR HARMONIC-FUNCTIONS WITH NONLINEAR BOUNDARY-CONDITIONS, Archive for Rational Mechanics and Analysis, 122(2), 1993, pp. 123-144
This paper is concerned with investigating the asymptotic behavior of
harmonic functions defined on a three-dimensional semi-infinite cylind
er, where homogeneous nonlinear boundary conditions are imposed on the
lateral surface of the cylinder. Such problems arise in the theory of
steady-state heat conduction. The classical Phragmen-Lindelof theorem
states that harmonic functions which vanish on the lateral surface of
the cylinder must either grow exponentially or decay exponentially wi
th distance from the finite end of the cylinder. Here we show that the
results are significantly different when the homogeneous Dirichlet bo
undary condition is replaced by the nonlinear heat-loss or heat-gain t
ype boundary condition. We show that polynomial growth (or deca ) or e
xponential growth (or decay) may occur, depending on the form of the n
onlinearity. Explicit estimates for the growth or decay rates are obta
ined.