Rev.Acad.Canar.Cienc ., I, 147-153 (1990)
ANALYTIC CHARACTERIZATIONS OF ANNULI
M~ Isabel Marrero Rodr{guez
Departamento de Analisis Matematico, Universidad de La .Laguna,
38271 La La9una (Tenerife), Spain
ABSTRACT
Characterizations of annuli among plane domains with analytic boundary in
· terms of potential theory and of quadrature identities are obtained using
elementary techniques.
KEYWORDS: Cauchy problem, harmonic measure, Poisson equation, quadrature identity.
I. INTRODUCTION AND MAIN RESULT
Let G be a finitely-connected plane domain whose boundary r consists of two
or more pairwise disjoint, analytic closed curves. Denote by Yo the "outer"
boundary component, and write y1=f-y 0 • Also, if A0 , P0 and A1 , P1 represent
the area and perimeter of the domains enclosed by Yo and y 1 , respectively, then
set μ0 =A0 /P 0 , μ 1 = -Ai/P 1 • The aim of this note is to prove the following result.
Theorem 1: In the above notation and hypotheses, let further dA represent the
area element in G, and lets denote the arclength parameter on r. Then, the
following are equivalent:
(i) The Cauchy problem
6v: -1 in G,
v= -μ~ on Yo, v= -μ~ on y 1 •
av av
an -μo on Yo, an -μ1 on yl
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is solvable in G. Here, av
an denotes the outer normal derivative of v on r.
(ii) The quadrature identity
ff u dA
G
holds for every harmonic function u in G.
au ds an ( 1)
(iii) G is an annulus centered at the origin with outer radius 2μ 0 and inner
radius -2μ 1 •
We recall that a quadrature identity for a vector space F of functions
defined and integrable with respect to area measure in the plane domain 0 is
an identity of the type
ff f dA
G
L( f) , (2)
valid for all functions f in the given class F, where L is some linear functional
on F. See, e. g., [1), [2) and references therein for further orientation on
this vaste subject.
Another characterization of the annulus by a quadrature identity has been
obtained by Ave! [3). A condition analogous to the one in part (i) of Theorem
1, also requiring the solvability of certain (overdetermined) Cauchy problem
for a Poisson differential equation, has been shown to characterize the balls
in the Euclidean space :Rn by Serrin [4 ). The connection between quadrature
identities and differential equations has been pointed out by Shapiro [s) .
It should be remarked, however, that the techniques used in [3] and [s] are not
suitable to deal with quadrature identities ( 2) where the defining functional L
is supported on the boundary of 0, as it happens in (1). Further, the methods
chosen in our approach are simple and elementary.
In the next Section II we present an auxiliary result, needed in the proof
of Theorem 1, which will be deferred until the last Section III. Throughout the
rest of this note, r will always stand for the dist 'ir 1· e from the origin in a:.
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II. AN AUXILIARY LEMMA
The proof of Theorem 1 is based on the following observation due to
Weinberger (6], which we label as Lemma 2.
Lemma 2: Let the real-valued functions t, v satisfy the Poisson equations 6t:-1,
6v:-1 in the plane domain G. Then,
w = lgrad vj 2 + t
is subharmonic in G. Moreover, w is harmonic if, and only if,
for some real constant c.
Proof: Since
1 (6v) 2 ~ 2
6jgrad vj 2 ,
we have
6w
v = - .!. r 2 + c
4
6jgrad vl 2
- 1 ~ 0,
( 3)
(4)
(5)
(6)
thus proving that w is subharmonic in G. If w is harmonic then equality holds in
(6). This leads to an equality also in (4) and (5). As 6v:-1, it follows that
- l
) and
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in G. Henc~, v= - ~ r 2 up to an additive real constant. The converse, that w
is harmonic when v has the form (3) and At:-1, is clear.
III. PROOF OF THEOREM 1
We begin by noticing that if (iii) holds then (3), with c=O, is a solution
of the Cauchy problem (i). Thus, we only need to show that (i) is equivalent
to (ii), and that (i) implies (iii).
(i) implies (ii): Let v be as stated in (i), and let u be harmonic in G. Then,
from Green's formula it follows that
ff u dA = -ff u Av dA
G G
= -ff G v Au dA + fr v :~ ds - fr u :~ ds
: f v au ds - f u av ds
r an r an
= μ 0 f u ds + μ 1 J u ds - μ ~ f : ~ ds - μff
lo li lo li
au
an ds .
This establishes (ii).
(ii) implies (i): Set t=u 0 - ~ r 2 , where u0 is the solution of Dirichlet's
problem in G with boundary data 41 r 2
• Note that t satisfies the Poisson
equation At:-1 in G and is zero on r. Next, let u be an arbitrary Cm function
on r, and denote also by u its harmonic extension into G. By Green's Formula,
where w0 , w1 are the harmonic measures of y 0 , y 1 , respectively (thatis,w
1
isthe
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solution of Dirichlet's problem in G with boundary data 1 on yi, 0 on r-yi (i=O,
1)). Now, being u arbitrary, we conclude that
The function v = t-μ~w 0 - μ~w 1 is thus a solution of the Cauchy problem (i).
(i) implies (iii): It suffices to show that v has the form (3) for some real
constant c. This will follow at once from Leltlfla 2 if we can prove that
w = lgrad vl 2 + t
is harmonic in G, where t is as described above. With this purpose we define
The function h is harmonic in G and has the same boundary values as w. Since w is
subharmonic (Lerrma 2), by the Maximum Principle either
w < h in G (7)
or
w - h in G. (8)
Nonoccurrence of (7) would complete the proof of Theorem 1. But (7) cannot hold,
since
JJ w dA
G
JJ h dA,
G
as we now proceed to show. Indeed, observing that
formula, the first member of (9) is computed to be
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(9)
t=v+h and using Green's
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ds + ff h dA
G
Another application of Green's Formula then yields
But, on the other hand,
f v r
av ds an
Insertion of (11) and <12) into (10) proves (9) and the Theorem.
REFERENCES
(10)
( 11)
( 12)
( 1 J M. Sakai: "Quadrature Domains" , Lecture Notes in Mathematics no. 934 ,
Springer-Verlag, Berlin, 1982.
(2] H. S. Shapiro: in C. A. Berenstein (Ed.), "Complex Analysis", Vol. 1, Lecture
Notes in Mathematics no. 1275, Springer-Verlag, Berlin, 1987, pp. 287-331.
[3] Y. Avci: Characterization of shell domains by quadrature identi ties. J. London
Math. Soc. (2), ~. 123-128 (1981).
(4] J. Serrin: A s ymmetry problem in potential theory· Arch. Rat. Mech. Anal . , 43,
304 - 318 (1971).
(s] H. S. Shapiro: in P.L. Butzer et al. (Eds.), " Armi versary Volume on
Approximation Theory and Functional Ana ! y si s" , Birkhauser Ver lag, Basel, 1984,
pp. 335 -354. ! 52
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(6) H. F. Weinberger: Remark on the preceding paper of Serrin. Arch. Rat. Mech.
Anal., 43, 319-320 (1971).
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