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

The elliptic system of partial differential equations of the thermistor problem is linearized to obtain the system
 \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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