Phragménlindelöf type results for harmonic functions with nonlinear boundary conditions
 C. O. Horgan,
 L. E. Payne
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Abstract
This paper is concerned with investigating the asymptotic behavior of harmonic functions defined on a threedimensional semiinfinite cylinder, where homogeneous nonlinear boundary conditions are imposed on the lateral surface of the cylinder. Such problems arise in the theory of steadystate heat conduction. The classical PhragménLindelöf 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 heatloss or heatgain type boundary condition. We show that polynomial growth (or decay) or exponential growth (or decay) may occur, depending on the form of the nonlinearity. Explicit estimates for the growth or decay rates are obtained.
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 Title
 Phragménlindelöf type results for harmonic functions with nonlinear boundary conditions
 Journal

Archive for Rational Mechanics and Analysis
Volume 122, Issue 2 , pp 123144
 Cover Date
 19930601
 DOI
 10.1007/BF00378164
 Print ISSN
 00039527
 Online ISSN
 14320673
 Publisher
 SpringerVerlag
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 Authors

 C. O. Horgan ^{(1)} ^{(2)}
 L. E. Payne ^{(1)} ^{(2)}
 Author Affiliations

 1. Department of Applied Mathematics, University of Virginia, 22903, Charlottesville, Virginia
 2. Department of Mathematics, Cornell University, 14853, Ithaca, New York