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A free boundary problem for a nonlinear degenerate elliptic system modeling a thermistor. (English) Zbl 0780.35116

Motivated by the mathematical model of a thermistor (electric circuit breaker), the paper is concerned with the unique solvability of the following elliptic system \[ \nabla(\sigma(u)\nabla\varphi)=0, \qquad \nabla^ 2 u=-\sigma(u) |\nabla\varphi|^ 2 \] in a rectangle, where \(\varphi\) is the electric potential and \(u\) denotes the temperature. In contrast to other papers studying this problem the electric conductivity \(\sigma\) is assumed to be \[ \sigma(u)=1\text{ if } u<u^*, \qquad \sigma(u)=0 \text{ if } u\geq u^*, \] where \(u^*\) is a critical temperature. The problem is completed by boundary conditions for \(u\) and \(\varphi\).
The proof of the unique solvability is splitted into several parts. At first, the original problem is transformed into a problem for \(\varphi\) and an auxiliary function \(\psi=u+ 0.5\;\varphi^ 2\). Then, by means of a conformal mapping the problem is reduced to a problem for \(\psi\) only, where \(\psi\) is determined by a variational inequality. The last step consists in the reconstruction of \(\varphi\) and \(\psi\) as functions defined in the original rectangle. Moreover, the set where \(\sigma(u)=0\) in the rectangle (i.e. infinite resistance) is found to be an interval or the empty set dependent on the quantity of the voltage drop across the thermistor.
Finally, the results are extended to more general two-dimensional domains.

MSC:

35R35 Free boundary problems for PDEs
35J70 Degenerate elliptic equations
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35R05 PDEs with low regular coefficients and/or low regular data
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References:

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