2004-Fez conference on Differential Equations and Mechanics. Electron. J. Diff. Eqns., Conference 11, 2004, pp. 135-142.

On a nonlinear problem modelling states of thermal equilibrium of superconductors

Mohammed El khomssi

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.

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Mohammed El Khomssi
UFR MDA Faculty of Sciences and Technology
Fez, Morocco
email: elkhomssi@fstf.ma

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