Appl Math Optim 50:143–159 (2004)
2004 Springer-Verlag New York, LLC
Heat Conduction with Flux Condition on a Free Patch
Kenneth L. Kuttler
and Meir Shillor
Department of Mathematics, Brigham Young University,
Provo, UT 84602, USA
Department of Mathematics and Statistics, Oakland University,
Rochester, MI 48309, USA
Communicated by D. Kinderlehrer
Abstract. A new free boundary or free patch problem for the heat equation is
presented. In the problem a nonlinear heat ﬂux condition is prescribed on a free
portion of the boundary, the patch, the position of which depends on the solu-
tion. The existence of a weak solution is established using the theory of set-valued
Key Words. Free boundary, Free patch, Heat equation, Weak solution, Existence.
This work deals with a new free boundary or “free patch” problem for the heat equation, in
which a nonlinear heat source is prescribed over a freely moving portion of the boundary.
Such problems arise naturally in contact processes where frictional heat generation takes
place over the contact patch, the position of which is one of the unknowns of the problem.
We establish the existence of a weak solution for the problem.
Let ⊂ R
be a domain with boundary = ∂, and denote by n the outer normal
to on . Let =
, and we assume, for the sake of simplicity, that
a planar domain in the xy plane. Furthermore,
is the potential surface
containing the free patch
Next, we describe the free patch. Let A be a ﬁxed, bounded, closed, and convex
set in R
with Lipschitz boundary ∂ A and nonempty interior. Let (X, Y ) denote a ﬁxed
point in the interior of A, chosen conveniently as the origin or center of A. When the