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