PHRAGMEN-LINDELOF TYPE RESULTS FOR HARMONIC-FUNCTIONS WITH NONLINEAR BOUNDARY-CONDITIONS

This paper is concerned with investigating the asymptotic behavior of harmonic functions defined on a three-dimensional semi-infinite cylinder, 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 with distance from the finite end of the cylinder. Here we show that the results are significantly different when the homogeneous Dirichlet boundary condition is replaced by the nonlinear heat-loss or heat-gain type boundary condition. We show that polynomial growth (or deca ) or exponential growth (or decay) may occur, depending on the form of the nonlinearity. Explicit estimates for the growth or decay rates are obtained.