A local partial regularity theorem for weak solutions of degenerate elliptic equations and its application to the thermistor problem
Xiangsheng Xu · Differential and Integral Equations · 1999
A partial regularity theorem is established for weak solutions of elliptic equations of the form $\mbox{div}(A(y) abla\psi)=0$. 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 $\frac{\partial u}{\partial t}-\Delta u = \sigma(u)| abla \varphi|^2$, $\mbox{div}(\sigma(u) abla\varphi)=0$ in the case where $\sigma$ may decay exponentially.