Let 1) be a region of connectivity n in the z-plane which contains the point of ... known as the Fredholm eigenvalues of C. They are all real, satisfy I~t I > 1, and ..... We now see that when D is a simply connected region, we can choose D* to.
FREDHOLM EIGENVALUES AND QUASICONFORMAL MAPPING BY GEORGE S P R I N G E R University of Kansas (1)
1. Introduction Let 1) be a region of connectivity n in the z-plane which contains the point of infinity and whose boundary C consists of n smooth J o r d a n curves C1, C2,..., Cn. Each curve Cr is the boundary of a bounded simply connected region Dj and we write D = (J~_ID s. The Neumann-Poincard integral equation is
/ (s) = ~ fc K(s, t) / (t) dr,
(1)
C, z(s) is a parametric repre~/~nt represents differentiation in the direc-
where s and t represent the arc length parameter on sentation of C in terms of its arc length,
z(t), and
tion of the inward normal at
a
K(s, t) = ~
1 log [z(s) -
(2)
z(t) ["
This integral equation plays an important role in potential theory and conformal mapping.
I t can be solved b y iteration and the Neumann-Liouville series so obtained
converges like a geometric series whose ratio is 1/I/t [ where ~t is the lowest eigenvalue of (1) whose absolute value is greater than one. known as the Fredholm eigenvalues of C.
The eigenvalues of (1) are
They are all real, satisfy I~t I ~> 1, and those
for which 121 > 1 lie symmetrically about the origin. Those of modulus one are referred to as the trivial eigenvalues.
I n order to have an estimate for the rate of conver-
gence of the Nenmann-Liouville series, it has been an important problem to estimate from below the lowest non-trivial positive Fredholm eigenvalue, which will be denoted by ~t in what follows. (1) This work was supported by the National Science Foundation.
122
G. SPRINGER
Ahlfors [1] showed how quasiconformal mapping leads to a very practical method for obtaining such estimates when C is the boundary of a simply connected region/~. In particular, he showed that if /~ admits a quasiconformal reflection [3] of maximal eccentricity k, i.e., a quasiconformal mapping / of /~ onto its exterior D which leaves C pointwise fixed and for which [/zl < k[/~], then 2/> 1/k. Since a quasiconformal reflection is not possible for multiply connected regions in the plane, Royden [10] embedded the given region /~ in a compact Riemann surface oil which the exterior of / ) had the same topological structure as /).
An in-
tegral equation analogous to (1) is then studied and its lowest non-trivial eigenvalue can be estimated by using a quasieonformal relfection. However the kernel K is no longer the Diriehlet kernel (2) but involves the Green's function of the Riemann surface which generally is unknown. Returning to simply connected domains, Warschawski [15, 16] showed how eigenvalue estimates can be obtained for a domain which is "close" to a domain for which such estimates are known, for example, for "nearly-circular" or "nearly-convex" domains.
Schiller [11] used variational methods to obtain such estimates for simply
connected regions and similar methods are used in [12, 13] to obtain estimates for c~rtain multiply connected regions. In this paper, a generalization of the Ahlfors method is presented which is applicable to multiply connected regions, and which gives a practical method for obtaining estimates for the lowest non-trivial positive eigenvalue 2 for the Neumann-Poinear~ equation (1). Let /-) be a domain of connectivity n containing co and having Jordan curves C1, C 2. . . . . Cn as its boundary.
Each Ck is the boundary of a simply connected
bounded domain D~, and D = I.J~=l Dk, C = I.J~=l Ck. We shall prove the following theorem. THEOICEM 1. Let ~(z) be a quasicon/ormal homeomorphism o/ the whole z-plane onto the whole $-plane with $(oo)= o~ which is K-quasieon/ormal in D and M-quasicon/ormal in ~ .
Let ~ map the curve system C onto a curve system C*.
We shall as-
sume that the Jordan curves in C ang C* have continuous curvature. 1 / 2 and 2* denote the Fredholm eigenvalue o/ C and C* respectively, then
1 2+1 2"+1 K _~,+1 KM2-1 < X ~ _ 1 ~< M ~ I .
(3)
Two curve systems C and C* are called quasiconformally equivalent if there is a quasiconformal homeomorphism of the whole plane which takes C onto C*. We
FI~EDHOLM E I G E N V A L U E S AND QUASICONFORMAL MAPPING
123
denote b y ~ * a class of canonical domains for domains of connectivity n, and assume that each domain /)*E ~* is bounded by a curve system C* for which 2*> I.
Let
:~* denote the class of curve systems C* which are boundaries of domains / ) * e ~*. We then have: THEOREM 2.
The conformal mapping o] ~) onto a domain D*eO* can be ex-
tended to a quasicon]ormal homeomorphism o] the whole plane if, and only i], 2 > 1. COROLLARY. The curve system C is quasiconformally equivalent to a curve system
C* E ~* i/, and only i], 2 > 1. In particular, C is quasicon]ormaUy equivalen$ to a system o/ circles if, and only if, 2 > 1. Finally the cross-ratio condition given by Ahlfors in [3] can be extended to curve systems C consisting of n-Jordan curves. We shall prove THEOR]~M 3. A curve system C i8 quasiconformally equivalent to a system of
circles i], and only if, there is a constant A such that (P1 P2 "P~ Pa)/ (P1Pa "P4 P~)
1
f f S , (v~*)~dw"
(20)
Mffo.(Va*),d,:
If we now take for ~* the harmonic conjugate ~* of the double-layer potential with density on C* a F r e d h o l m eigenfunetion corresponding to - 2 " ,
we can define
/l(Z)=9*(~(z)) zeC. W e next solve the Dirichlet problem in ~) and D for functions ~1 a n d h 1 respectively, b o t h of which assume the b o u n d a r y values /1 on C.
The function )bl m a y n o t have
a single-valued conjugate, b u t b y adding a suitable linear combination of harmonic measures to h 1, we get an admissible fucntion gl in ~ and b y adding suitable constants to h 1 in each c o m p o n e n t of D, we get a function gl in D such t h a t gl, gl form an admissible pair for C. ponent
of Ca, the
Since gl and [1 differ b y constants on each com-
chain of operations going from ( 1 5 ) t o ( 2 0 ) s t a r t i n g with ~ = g l
and h = gl lead to the extremal admissible pair ~ * = ~* a n d g * = g*. This allows us to write
). -]- 1
(Vgl)2d~'z
- 1
(Vgl)2d~:z
'ff
M
9 (Vg~)~dv~
1 )~* + 1 (21)
(Vg*) 2 d ~ $
Thus we have proved (3) for the case in which K = 1.
FREDHOLM EIGENVALUES
AND QUASICONFORMAL MAPPING
129
5. Factorization of quasieonformal mappings To complete the proof of Theorem 1, we shall use the following lemma: L]~MMA 1. Every homeomorphism / o/ the plane which is K-quasicon/ormal in D
and M-quasicon/ormal in D can be /actored to the composition o/two mappings /= hog where g is con]ormal in ~ and K-quasicon/ormal in D and h is con/ormal in g(D) and M-quasieon/ormal in g([)). Since / is quasiconformal, it has generalized derivatives /z and /~ which satisfy /z=/~/~ where HlaH~ a > 1. Thus the region / ) is the conformal
Da represent the interior of the ellipse Ca and Do the exterior of the ellipse Cb. The function ~ = z + 1/z
image of the annulus a < I z [ < b under the mapping ~ = z + 1/z. Let
also gives us a conformal mapping o f [ z ] > b onto Db and the extension to ] z l < a
Da with maximal eccentricity 1/a 2. The Fredholm eigenvalue 2 for the annulus a < I z] < b is 2 = (b/a)2; cf. [13]. Then
defined by (23) gives us a mapping of ]zl< a onto Theorem 1 gives us the estimate 1 b2 2"+1 1 +a-~a~+ 1 2"-1 ~ R.
Such a mapping is given by
Ig(~O)re+~
for r < ~
[ re+~
for r/> R.
Using the facts that ~r --.
r
~r
z
~z 2 z' ~
~0
2 r' ~z
=
1
~0 - - - 1
2 iz' ~
2iX'
]F~/Fz ]. I n order to express the results in geomeg'(O)/g(O)=tan v(O), where v(0) is the angle between the radius vector from 0 to g(O)e~~ and the normal vector to C at g(O)e+~ We have we can compute expressions for trical terms, let us observe that
tan 2 ~(0) 4 + tan ~ ~(0) 1-- ~
IF+/F+ I =
for r
R
0
for r > R .
134
O.
SPRINGER
Let us introduce the following notation: a =g.l.b.
~
fl= 1.u.b. ~
- Q
We can then show that if ~ is chosen such that a < 1 ([ z [ = ~ lies inside of C) 72 4--+-r~
for
]$':/Fzl 2
/1 (C lies inside of ]z I = ~), then 72 ~ _ 7~
for r
e~" would hold for a n y points P Eft and Q E ~.
to
Thus a and fl
would be separated b y an annulus whose radii have ratio e~'~. The e x t r e m a l distance between the
two
circles of such an annulus is 1, so dDk (~, fl)> 1, a contradiction.
Thus we m u s t have P1 P~ ~
