A local partial regularity theorem for weak solutions of degenerate elliptic equations and its application to the thermistor problem.
type: Item
instance of: scholarly article
title: A local partial regularity theorem for weak solutions of degenerate elliptic equations and its application to the thermistor problem.
zbMATH Open document ID: 1064.35063
author: Xiangsheng Xu
published in: Differential and Integral Equations
publication date: 2003-01-26T00:00:00Z
review text: The author is inte\-res\-ted in the ther\-mis\-tor problem, i.e. in the ini\-tial-boun\-da\-ry va\-lue problem \[ \begin{aligned} \begin{split} \frac {\partial u}{\partial t} - \triangle u = \sigma (u)| \nabla \phi | ^2 &\quad \text{on}\quad \Omega _T = \Omega \times (0,T),\\ \text{div}\;(\sigma (u) \nabla \phi ) = 0 &\quad \text{on}\quad \Omega _T,\\ u(x,t) = u_0 (x,t) &\quad \text{on}\quad \partial _p\Omega _T = (\partial \Omega \times (0,T))\cup (\Omega \times \{0\}),\\ \phi (x,t) = \phi _0(x,t) &\quad \text{on}\quad S_T =\partial \Omega \times (0,T), \end{split} \end{aligned} \] in case when \(\sigma \) can decay to zero exponentially. The difficulty lies in the degeneracy of the second equation together with the quadratic nonlinearity in the first equation. Under natural smoothness assumptions on initial and boundary data the existence and regularity of a weak solution \(u,\phi \) is established. The result is obtained by using a partial regularity theorem for degenerate elliptic equations suited for the above described case.
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35D10
Mathematics Subject Classification ID: 78A55
Mathematics Subject Classification ID: 35K65
zbMATH DE Number: 1858737
zbMATH Keywords: thermistor problem
zbMATH Keywords: degenerate elliptic equations
zbMATH Keywords: weak solutions
zbMATH Keywords: partial regularity
reviewed by: J. Stará
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1858737