Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 38, pp. 1-14.

Title: On the dimension of the kernel of the linearized thermistor operator
        
Author: Giovanni Cimatti (Univ. of Pisa, Italy)
        
Abstract:
 The elliptic system of partial differential equations of the
 thermistor problem is linearized to obtain the system
 $$\displaylines{
 \nabla\cdot(\sigma(\bar u)\nabla\Phi+\sigma'(\bar u)U\nabla\bar\varphi)=0
 \quad\text{in }\Omega,\quad  \Phi=0\quad\hbox{on }\Gamma\cr
 \Delta U+\sigma'(\bar u)|\nabla\bar\varphi|^2 U+2\sigma(\bar u)\nabla\bar
 \varphi \cdot\nabla\Phi=0\quad
 \hbox{in }\Omega, \quad U=0\quad\hbox{on } \Gamma.
 }$$
 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 01, 2013.
Math Subject Classifications: 35B15, 35J66.
Key Words: Elliptic system; thermistor problem; existence; 
           uniqueness of solutions.