A local partial regularity theorem for weak solutions of degenerate elliptic equations and its application to the thermistor problem.
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.
scientific article; zbMATH DE number 1858737