Electron. J. Diff. Equ., Vol. 2013 (2013), No. 38, pp. 1-14.

On the dimension of the kernel of the linearized thermistor operator

Giovanni Cimatti

Abstract:
The elliptic system of partial differential equations of the thermistor problem is linearized to obtain the system
$$\displaylines{ \nabla\cdot(\sigma(\bar u)\nabla\Phi+\sigma'(\bar u)U\nabla\bar\varphi)=0 \quad\hbox{in }\Omega,\quad \Phi=0\quad\hbox{on }\Gamma\cr \Delta U+\sigma'(\bar u)|\nabla\bar\varphi|^2 U+2\sigma(\bar u)\nabla\bar \varphi \cdot\nabla\Phi=0\quad \hbox{in }\Omega, \quad U=0\quad\hbox{on } \Gamma. }$$
We study the existence of nontrivial solutions for this linear boundary-value problem, which is useful in the study of the thermistor problem.

Submitted September 4, 2012. Published February 1, 2013.
Math Subject Classifications: 35B15, 35J66.
Key Words: Elliptic system; thermistor problem; existence; uniqueness of solutions.

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Giovanni Cimatti
Department of Matematics, University of Pisa
Largo Bruno Pontecorvo 5, 56127 Pisa, Italy
email: cimatti@dm.unipi.it

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