Nonlinear elliptic systems with variable boundary data
type: Item
instance of: scholarly article
title: Nonlinear elliptic systems with variable boundary data
zbMATH Open document ID: 1087.35031
DOI: 10.1016/S0362-546X(02)00179-7
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-03-11T00:00:00Z
review text: The authors consider systems of elliptic nonlinear partial differential equations \[ -\Delta u(x)=G_u(x,u(x),v(x)), \] \[ \Delta v(x)=G_v(x,u(x),v(x)), \] \[ u=g \text{ on } \Gamma, \quad u=h \text{ on } \Gamma, \] where \(g(\cdot)\in H^{1/2}(\Gamma, \mathbb R^N)\), \(h(\cdot)\in H^{1/2}(\Gamma,\mathbb R^M)\), \(u(\cdot)\in H^{1}(\Omega, \mathbb R^N)\), \(v(\cdot)\in H^{1}(\Omega, \mathbb R^M)\), \(G\) is a scalar function defined on the set \(\Omega\times \mathbb R^N\times \mathbb R^M\), \(G_u=(G_{u^1}),\dots,G_{u^N})\), \(G_v=(G_{v^1}),\dots,G_{v^M})\), \(\Omega\subset \mathbb R^n\), \(n\geq2\), \(\Gamma=\partial \Omega\). The question of the continuous dependence on boundary data \((g,h)\in W\) of solutions (in the weak sense) in the space \(H^1(\Omega, \mathbb R^N)\times H^1(\Omega, \mathbb R^M)\) is investigated. Under some assumption this question is replaced with the question of the continuous dependence on the boundary data of saddle points of the functional of action \[ F(u,v)=\int_{\Omega} \left(\frac{1}{2}\nabla v(x)^2)-\frac{1}{2}\nabla u(x)^2 +G(x,u(x),v(x))\right) dx \] for the considered system.
reviewed by: Josef Diblík
Mathematics Subject Classification ID: 35J55
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35J50
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 47J30
zbMATH DE Number: 1880012
zbMATH Keywords: Nonlinear elliptic equations
zbMATH Keywords: Boundary value problems
zbMATH Keywords: Continuous dependence on boundary data
zbMATH Keywords: Stability
author: Dorota Bors
author: Stanislaw Walczak
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1880012