View source for Fredholm eigenvalue of a Jordan curve

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Let $C$ be a smooth [[Jordan curve|Jordan curve]] (of class $C ^ { 3 }$) in the complex $z = ( x + i y )$-plane, $G$ its interior, $G ^ { * }$ its exterior. Then, let

\begin{equation*} K ( x , t ) = - \frac { 1 } { \pi } \frac { \partial } { \partial n _ { t } } \operatorname { log } | z - t | , z , t \in C, \end{equation*}

be the corresponding classical Neumann kernel with $n_t$ the interior normal. The Fredholm eigenvalue $\lambda$ of $C$ is the smallest eigenvalue $&gt; 1$ of this kernel.

This eigenvalue plays an important role for the speed of the successive approximation solution of several problems, such as the integral equation with this kernel corresponding to the [[Dirichlet problem|Dirichlet problem]] and several integral equations to construct the conformal Riemann mapping function of $G$ [[#References|[a1]]].

For arbitrary Jordan curves $C$, there is the following characterization of the Fredholm eigenvalue $\lambda$ [[#References|[a4]]]:

\begin{equation*} \frac { 1 } { \lambda } = \operatorname { sup } \frac { | D ( h ) - D ^ { * } ( h ) | } { D ( h ) + D ^ { * } ( h ) }, \end{equation*}

where the supremum is over all functions $h$ that are continuous in the extended plane and harmonic in $G$ and in $G ^ { * }$, with corresponding Dirichlet integrals $D ( h )$ and $D ^ { * } ( h )$ (cf. also [[Dirichlet integral|Dirichlet integral]]).

One has: $1 \leq \lambda \leq \infty$, with $\lambda = \infty$ if and only if $C$ is a circle, and with $\lambda &gt; 1$ if and only if $C$ is a quasi-circle [[#References|[a4]]], [[#References|[a2]]]. In some sense, $\lambda$ is a measure for the deviation of $C$ from a circle. $\lambda$ is invariant under Möbius transformations (cf. also [[Fractional-linear mapping|Fractional-linear mapping]]).

The exact value of $\lambda$ is known for several special Jordan curves: e.g. ellipses, some Cassinians, triangles, regular $n$-gons, and rectangles close to a square.

There is also the following characterization of $\lambda$, using the Riemann mapping of $G ^ { * }$. Without loss of generality one may assume that $G ^ { * }$ is the image of $| \zeta | &gt; 1$ under a univalent conformal mapping (cf. also [[Conformal mapping|Conformal mapping]]) $z ( \zeta )$ of the form

\begin{equation*} z ( \zeta ) = \zeta + \frac { a _ { 1 } } { \zeta } + \frac { a _ { 2 } } { \zeta ^ { 2 } } + \ldots \end{equation*}

One can then calculate the so-called Grunsky coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014036.png"/> in the development

\begin{equation*} \operatorname { log } \frac { z ( \zeta ) - z ( \zeta ^ { \prime } ) } { \zeta - \zeta ^ { \prime } } = - \sum _ { k , l = 1 } ^ { \infty } a _ { k l } \zeta ^ { - k } \zeta ^ { \prime - l }, \end{equation*}

\begin{equation*} | \zeta | &gt; 1 , | \zeta ^ { \prime } | &gt; 1. \end{equation*}

Then

<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014039.png"/></td> </tr></table>

where the supremum is taken over all complex numbers $x _ { k }$ with $\sum _ { k = 1 } ^ { \infty } | x _ { k } | ^ { 2 } / k = 1$. This gives also a procedure to evaluate $\lambda$ numerically [[#References|[a2]]].

Hence one obtains simple upper estimates for $\lambda$, of course. There are many other such estimates in which the mapping $z ( \zeta )$ is involved [[#References|[a2]]]. As a simple consequence, there is the following very useful inequality for Jordan curves $C$ with corners [[#References|[a2]]]:

\begin{equation*} \frac { 1 } { \lambda } \geq | 1 - \alpha |. \end{equation*}

Here, $\alpha \pi$ denotes the angle at the corner.

For large $\lambda$ one finds [[#References|[a2]]] that $C$ must be contained in an annulus with radii $r$ and $r ( 1 + 2.78 / \lambda )$ such that $C$ separates the boundary circles with these radii. However, in the other direction, $\lambda$ can be close to $1$ even though $C$ lies, in the same manner, in an annulus for which the quotient of the radii is arbitrarily close to $1$.

There are also several lower estimates from M. Schiffer and others for $\lambda$ [[#References|[a1]]], [[#References|[a2]]], [[#References|[a4]]]. If, for example, $C$ is the image of $| \zeta | = 1$ under a univalent conformal mapping of an annulus $r &lt; | \zeta | &lt; R$ ($r &lt; 1 &lt; R$), then

\begin{equation*} \lambda \geq \frac { r ^ { 2 } + R ^ { 2 } } { 1 + ( r R ) ^ { 2 } }. \end{equation*}

L.V. Ahlfors noted a remarkable interaction between the theory of Fredholm eigenvalues and the theory of [[Quasi-conformal mapping|quasi-conformal mapping]]: If there is a $Q$-quasi-conformal reflection at $C$ (i.e., a sense-reversing $Q$-quasi-conformal mapping of the extended plane which leaves $C$ pointwise fixed), then [[#References|[a2]]], [[#References|[a4]]]

\begin{equation*} \lambda \geq \frac { Q + 1 } { Q - 1 }. \end{equation*}

The question of equality gives rise to interesting connections with the theory of extremal quasi-conformal mappings (connected with the names of O. Teichmüller, K. Strebel, E. Reich; cf. [[#References|[a2]]]).

From this Ahlfors inequality one obtains almost immediately [[#References|[a2]]], [[#References|[a4]]]:

\begin{equation*} \frac { 1 } { \lambda } \leq \operatorname { max } _ { \varphi } | \operatorname { cos } \alpha ( \varphi ) | \end{equation*}

if $C$ is smooth and starlike with respect to the interior point $z = 0$, where $\alpha ( \varphi )$ denotes the angle between the ray $\operatorname{arg} z = \varphi$ and the tangent at the point of $C$ with $\varphi$.

For a theory of Fredholm eigenvalues for multiply-connected domains, see [[#References|[a3]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  D. Gaier,   "Konstruktive Methoden der konformen Abbildung" , Springer  (1964)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  R. Kühnau,   "Möglichst konforme Spiegelung an einer Jordankurve"  ''Jahresber. Deutsch. Math. Ver.'' , '''90'''  (1988)  pp. 90–109</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  M. Schiffer,   "Fredholm eigenvalues of multiply-connected domains"  ''Pacific J. Math.'' , '''9'''  (1959)  pp. 211–269</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  G. Schober,   "Estimates for Fredholm eigenvalues based on quasiconformal mapping" , ''Lecture Notes Math.'' , '''333''' , Springer  (1973)  pp. 211–217</td></tr></table>