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International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 1, Pages 113-116

A temperature control problem

Department of Mathematics, University of Kentucky, Lexington 40506, KY, USA

Received 10 April 1983

Copyright © 1984 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In an open bounded region of n-space occupied by a homogeneous and isotropic medium, we control the temperature through the boundary. The normal derivative of the temperature (which measures the appropriate heat flux) is restricted to be nonnegative. This gives rise to a free boundary in space-time separating the areas of positive and zero heat flux. Under a natural monotonicity condition, the free boundary is the graph of a function of space. This function is shown to be locally Lipschitz. Moreover for n=2 the time derivative of the temperature is proven to be continuous across the free boundary.