CHAPTER 1
Introduction and Main Results
Let μ be a finite positive Borel measure in the Euclidean plane
R2
with compact
support and let λ be the two-dimensional Lebesgue measure. If a bounded open
set Ω in
R2
satisfies
μ(R2
\ Ω) = 0 and
sdμ
Ω
sdλ
for every subharmonic function s in Ω, which is integrable on Ω with respect to λ,
we call Ω a quadrature domain of μ. For the fundamental properties of quadrature
domains and their developments, see [D], [S1], [Sp] and [EGKP].
In this paper, we discuss the case in which μ is of the form
λ|Ω(0) + t · δp0 ,
where λ|Ω(0) denotes the restriction of λ onto a given bounded open set Ω(0), δp0
denotes the Dirac measure at a fixed point p0 in Ω(0) and t is a small positive
number. Namely, we discuss the bounded open set Ω(t) Ω(0) for small t 0
such that
Ω(0)
sdλ + t · s(p0)
Ω(t)
sdλ
for every subharmonic and integrable function s in Ω(t). In other words, we discuss
small modifications of quadrature domains. We focus on the case of Ω(0) having
a corner on the boundary and give a detailed discussion about the shape of Ω(t)
around the corner. This small modification is one of the subjects in pure mathe-
matics, but it has a model in hydrodynamics: the Hele-Shaw flow. We shall adopt
several names from the model and describe our results.
A flow which is produced by the injection of fluid into the narrow gap be-
tween two parallel planes is called a Hele-Shaw flow(see [Hl-Sw], [L, p.582], [Ga],
[Pl-Ko], [VKu], [R] and [S1]). A mathematical description of this flow is as fol-
lows: Let Ω(0) be a bounded connected open set in the plane and let p0 be a point
in Ω(0). Here we define Ω(0) and p0 as the projection of the averaged initial blob
of fluid and the injection point of fluid into one of the two parallel planes, respec-
tively. The Hele-Shaw flow {Ω(t)}t0 comprises the monotone increasing family of
bounded connected open sets Ω(t) such that
(1.1)
1

∂G(z, p0, Ω(t))
∂nz
=
1
∂t
∂nz
for every t 0 and every point z on the boundary ∂Ω(t) of Ω(t), where G(z, p0, Ω(t))
denotes the Green function (of the Dirichlet problem for the Laplace operator) for
Ω(t) with pole at p0, ∂/∂nz denotes the outer normal derivative at z ∂Ω(t),
t = t(z) denotes the time determined by z ∂Ω(t), and we assume that ∂Ω(t) is
smooth for every t 0 and t(z) is also smooth as a function of z Ω(0). The
1
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