Chapter 1
An Introductory
Problem
1.1. Introduction and heuristic considerations
In this part we consider free boundary problems of the following type.
In the ball B\ B\ (0) we have a continuous function u satisfying
(a) Au = 0 in n+(u) := {u 0} and in Q-(u) := {u 0}°,
(b) the flux balance
(1.1) G(u+,u-) = 0
across F(u) :=
d£l+(u)
f) B\, the free boundary.
In (1.1), u+ and u~ denote the normal derivatives in the inward direction
to
Q+(u)
and Cl~(u), respectively, so that u^ are both nonnegative.
The simplest example of this type of problems arises in the minimization
of the variational integral
(1.2) J0(u)= f
{\Vu\2
+ X{u0}) dx
JBi
that appears in many applications (e.g., in jet flows, see [AC], [ACFl],
[ACF2], [ACF3]).
Suppose u is a local minimizer and assume that the free boundary is
differentiable (say) at the origin.
Since
ux(x)
= -r^(Ax)
3
http://dx.doi.org/10.1090/gsm/068/01
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