Phragmén-Lindelöf type results for harmonic functions with nonlinear boundary conditions
type: Item
instance of: scholarly article
title: Phragmén-Lindelöf type results for harmonic functions with nonlinear boundary conditions
zbMATH Open document ID: 0790.31002
DOI: 10.1007/BF00378164
author: Cornelius O. Horgan
author: L. E. Payne
published in: Archive for Rational Mechanics and Analysis
publication date: 1993-09-07T00:00:00Z
review text: The subject of this study are harmonic functions \(u\) in a semi-infinite cylinder \(\{(x_ 1,x_ 2,x_ 3):(x_ 1,x_ 2) \in D\}\) with prescribed Dirichlet and Neumann conditions on \(D\) and nonlinear boundary conditions \(\partial u/ \partial n+f(u)=0\) on \(\partial D \times \mathbb{R}^ +\). No conditions are given at infinity. It is shown that the \(L_ 2\)- norm \[ \varphi(z)=\int_ Du^ 2 (x_ 1,x_ 2,z) dx_ 1 dx_ 2 \] decays or grows exponentially or polynomially according to the growth of the nonlinearity \(f\). The proof is based on a differential inequality for \(\varphi (z)\). This inequality is obtained by means of elementary arguments, however it requires a great skill and knowledge of the structure of harmonic functions.
reviewed by: Catherine Bandle
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 410078
zbMATH Keywords: \(L_ 2\)-norm
zbMATH Keywords: exponential decay
zbMATH Keywords: exponential growth
zbMATH Keywords: polynomial decay
zbMATH Keywords: polynomial growth
zbMATH Keywords: harmonic functions
zbMATH Keywords: Dirichlet and Neumann conditions
zbMATH Keywords: nonlinear boundary conditions
zbMATH Keywords: differential inequality
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 410078