## Differential and Integral Equations

### A local partial regularity theorem for weak solutions of degenerate elliptic equations and its application to the thermistor problem

Xiangsheng Xu

#### Abstract

A partial regularity theorem is established for weak solutions of elliptic equations of the form $\mbox{div}(A(y)\nabla\psi)=0$. Here we allow the possibility that the eigenvalues of $A(y)$ are not bounded away from $0$ below. This result is then used to prove an everywhere regularity theorem for weak solutions of the initial- boundary-value problem for the system $\frac{\partial u}{\partial t}-\Delta u = \sigma(u)|\nabla \varphi|^2$, $\mbox{div}(\sigma(u) \nabla\varphi)=0$ in the case where $\sigma$ may decay exponentially.

#### Article information

Source
Differential Integral Equations, Volume 12, Number 1 (1999), 83-100.

Dates
First available in Project Euclid: 29 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1367266995

Mathematical Reviews number (MathSciNet)
MR1668541

Zentralblatt MATH identifier
1064.35063

Subjects
Primary: 35J70: Degenerate elliptic equations
Secondary: 35D10 78A55: Technical applications

#### Citation

Xu, Xiangsheng. A local partial regularity theorem for weak solutions of degenerate elliptic equations and its application to the thermistor problem. Differential Integral Equations 12 (1999), no. 1, 83--100. https://projecteuclid.org/euclid.die/1367266995