Appl Math Optim 47:231–252 (2003)
2003 Springer-Verlag New York Inc.
A Free Boundary Problem Modelling the Electrolysis of Aluminium
and M. C. Mu˜niz
Institut f¨ur Angewandte Mathematik, Universit¨at Z¨urich,
8057 Z¨urich, Switzerland
Departamento de Matem´atica Aplicada, Universidade de Santiago de Compostela,
15706 Santiago, Spain
Communicated by D. Kinderlehrer
Abstract. The goal of this note is to show existence and uniqueness for a stationary
one-phase Stefan problem with source at the free boundary modelling the electrolysis
Key Words. Free boundary, Aluminium electrolysis, Existence.
AMS Classiﬁcation. 35J20.
We study here a free boundary problem arising in the thermal modelling of an electrolytic
cell. Starting from the problem as formulated in , we are looking for a pair (T , q),
where T is the temperature and q is the characteristic function of the liquid set, which
is the solution of a nonlinear problem introduced in Section 2.
In Section 3 we prove the existence of (T, q) assuming that the thermal conductivity
depends on the spatial variable and on the temperature, and in doing so we generalize
and simplify the results of  and .
This work has been partially supported by the Swiss National Science Foundation under Contract
#20-58856.99 and by Xunta de Galicia (PGIDT00PXI20701PR)