Fredholm eigenvalues and quasiconformal mapping - Project Euclid

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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~ ~