2004-Fez conference on Differential Equations and Mechanics.
Electronic Journal of Differential Equations,
Conference 11, 2004, pp. 135-142.
Title: On a nonlinear problem modelling states of thermal
equilibrium of superconductors
Author: Mohammed El khomssi (Faculty of Sciences and Technology, Fez, Morocco)
Abstract:
Thermal equilibrium states of superconductors are governed
by the nonlinear problem
$$
\sum_{i=1}^{i=N}\frac{\partial }{\partial x_{i}}
\big(k(u) \frac{\partial u}{\partial x_{i}}\big)=\lambda
F(u) \quad \hbox{in } \Omega \,,
$$
with boundary condition $u=0$. Here the domain $\Omega $ is an open
subset of $\mathbb{R}^{N}$ with smooth boundary.
The field $u$ represents the thermal state, which we assume is in
$H_{0}^{1}( \Omega )$. The state $u=0$ models the superconductor's
state which is the unique physically meaningful solution.
In previous works, the superconductor domain is unidirectional
while in this paper we consider a domain with arbitrary
geometry. We obtain the following results:
A set of criteria that leads to uniqueness of a superconductor state,
a study of the existence of normal states and the number of them,
and optimal criteria when the geometric dimension is 1.
Published October 15, 2004.
Math Subject Classifications: 35J60, 34L30, 35Q99.
Key Words: Equilibrium states; nonlinear; thermal equilibrium; superconductors.