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.