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.