Existence results for a nonlinear elliptic system modeling a temperature dependent electrical resistor
type: Item
instance of: scholarly article
title: Existence results for a nonlinear elliptic system modeling a temperature dependent electrical resistor
zbMATH Open document ID: 0675.35039
DOI: 10.1007/BF01766151
author: Giovanni Cimatti
author: Giovanni Prodi
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1988-01-01T00:00:00Z
review text: The authors study the particular nonlinear elliptic system: \[ \nabla \cdot (\sigma (u)\nabla \phi)=0,\quad -\nabla (k \nabla u)=\sigma (u)| \nabla \phi |^ 2, \] for the temperature u and electric potential \(\phi\), modelling the steady state behaviour f a thermally sensitive resistor (\(\sigma\), k denote electrical and thermal conductivity). The boundary values of u, \(\phi\) are held constant on the smooth boundary of the body under consideration. We note that the system displays a quadratic dependence on the electric field \(\nabla \phi\). Based on the Leray-Schauder fixpoint principle and an approximation procedure existence of a weak solution is obtained.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 78A30
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35A35
Mathematics Subject Classification ID: 35Q99
zbMATH DE Number: 4105431
zbMATH Keywords: Joule effect
zbMATH Keywords: temperature
zbMATH Keywords: electric potential
zbMATH Keywords: steady state
zbMATH Keywords: thermally sensitive resistor
zbMATH Keywords: smooth boundary
zbMATH Keywords: quadratic dependence
zbMATH Keywords: Leray-Schauder fixpoint principle
zbMATH Keywords: approximation
zbMATH Keywords: existence
zbMATH Keywords: weak solution
MaRDI profile type: Publication
cites work: Q4748596
cites work: Q3037899
cites work: Q3715516
cites work: Q5187597
cites work: Linear and quasilinear elliptic equations
cites work: Q5334555
full work available at URL: https://doi.org/10.1007/bf01766151
OpenAlex ID: W1979511880
description: scientific article; zbMATH DE number 4105431