A nonlinear elliptic boundary value problem relevant in general relativity and in the theory of electrical heating of conductors

Abstract

The elliptic boundary value problem governing the steady electrical heating of a conductor of heat and electricity, the so-called thermistor problem,

$$\begin{aligned}&{\nabla }\cdot ({\sigma }(u){\nabla }\phi )=0\ {\quad \hbox {in}\ {\Omega }}\quad \phi =\phi _b\ {\quad \hbox {on}\ {\Gamma }}\\&{\nabla }\cdot ({\kappa }(u){\nabla }u)=-{\sigma }(u)|{\nabla }\phi |^2\ {\quad \hbox {in}\ {\Omega }}\quad u=0\ {\quad \hbox {on}\ {\Gamma }}, \end{aligned}$$

where ${\sigma }(u)$ is the temperature dependent electric conductivity and ${\kappa }(u)$ the thermal conductivity, admits a reinterpretation in the framework of general relativity if we choose ${\sigma }(u)=e^u$, ${\kappa }(u)=1$ and, in addition, ${\Omega }$ is a domain of ${\mathbf{R}^3}$ axially symmetric whereas the function $\phi _b$, in a cylindrical coordinate system ${\rho },z,{\varphi }$, is independent of ${\varphi }$. The same analytical methods relevant in the thermistor problem can be used in this new context.