## Differential and Integral Equations

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

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

#### 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