Hidden symmetry of fully nonlinear boundary conditions in elliptic equations: Global bifurcation and nodal structure
type: Item
instance of: scholarly article
title: Hidden symmetry of fully nonlinear boundary conditions in elliptic equations: Global bifurcation and nodal structure
zbMATH Open document ID: 0760.35010
DOI: 10.1007/BF03323073
author: Timothy J. Healey
author: Hansjörg Kielhöfer
published in: Results in Mathematics
publication date: 1992-09-27T00:00:00Z
review text: The nonlinear elliptic boundary value problem under consideration has the form \[ -\nabla\cdot{\mathbf g}(\nabla u,u)=f(\nabla u,u,\lambda)\quad\text{in }\Omega,\quad{\mathbf g}(\nabla u,u)\cdot{\mathbf n}|_{\partial\Omega}=0 \] in a rectangle \(\Omega\), where \(f,{\mathbf g}\) are prescribed functions, \({\mathbf n}\) denotes the unit external normal to \(\partial\Omega\), and \(\lambda\) corresponds to a control parameter in a steady-state diffusion process. The ``hidden symmetry'' of the boundary condition permits application of the technique used previously for analogous problems with linear boundary conditions, leading to generalizations of results of \textit{M. G. Crandall} and \textit{P. H. Rabinowitz} [J. Math. Mech. 19, 1083-1102 (1970; Zbl 0206.097)]. In particular, an existence proof is given for bifurcating solution branches along which nodal configurations of linearized eigenfunctions are preserved.
reviewed by: Charles A. Swanson
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 62156
zbMATH Keywords: steady-state diffusion process
MaRDI profile type: Publication
cites work: Q5606061
cites work: Symmetry and related properties via the maximum principle
cites work: Global Bifurcations and Continuation in the Presence of Symmetry with an Application to Solid Mechanics
cites work: Symmetry and nodal properties in the global bifurcation analysis of quasi-linear elliptic equations
cites work: Multiple eigenvalue bifurcation for Fredholm operators.
cites work: Q5533735
cites work: Some aspects of nonlinear eigenvalue problems
cites work: A symmetry problem in potential theory
description: scientific article; zbMATH DE number 62156