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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f1201401.png" /> be a smooth [[Jordan curve|Jordan curve]] (of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f1201402.png" />) in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f1201403.png" />-plane, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f1201404.png" /> its interior, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f1201405.png" /> its exterior. Then, let
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f1201406.png" /></td> </tr></table>
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be the corresponding classical Neumann kernel with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f1201407.png" /> the interior normal. The Fredholm eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f1201408.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f1201409.png" /> is the smallest eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014010.png" /> of this kernel.
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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
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014011.png" /> [[#References|[a1]]].
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\begin{equation*} K ( x , t ) = - \frac { 1 } { \pi } \frac { \partial } { \partial n _ { t } } \operatorname { log } | z - t | , z , t \in C, \end{equation*}
  
For arbitrary Jordan curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014012.png" />, there is the following characterization of the Fredholm eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014013.png" /> [[#References|[a4]]]:
+
be the corresponding classical Neumann kernel with $n_t$ the interior normal. The Fredholm eigenvalue $\lambda$ of $C$ is the smallest eigenvalue $> 1$ of this kernel.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014014.png" /></td> </tr></table>
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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]]].
  
where the supremum is over all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014015.png" /> that are continuous in the extended plane and harmonic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014016.png" /> and in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014017.png" />, with corresponding Dirichlet integrals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014019.png" /> (cf. also [[Dirichlet integral|Dirichlet integral]]).
+
For arbitrary Jordan curves $C$, there is the following characterization of the Fredholm eigenvalue $\lambda$ [[#References|[a4]]]:
  
One has: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014020.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014021.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014022.png" /> is a circle, and with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014023.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014024.png" /> is a quasi-circle [[#References|[a4]]], [[#References|[a2]]]. In some sense, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014025.png" /> is a measure for the deviation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014026.png" /> from a circle. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014027.png" /> is invariant under Möbius transformations (cf. also [[Fractional-linear mapping|Fractional-linear mapping]]).
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\begin{equation*} \frac { 1 } { \lambda } = \operatorname { sup } \frac { | D ( h ) - D ^ { * } ( h ) | } { D ( h ) + D ^ { * } ( h ) }, \end{equation*}
  
The exact value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014028.png" /> is known for several special Jordan curves: e.g. ellipses, some Cassinians, triangles, regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014029.png" />-gons, and rectangles close to a square.
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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]]).
  
There is also the following characterization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014030.png" />, using the Riemann mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014031.png" />. Without loss of generality one may assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014032.png" /> is the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014033.png" /> under a univalent conformal mapping (cf. also [[Conformal mapping|Conformal mapping]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014034.png" /> of the form
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One has: $1 \leq \lambda \leq \infty$, with $\lambda = \infty$ if and only if $C$ is a circle, and with $\lambda > 1$ 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]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014035.png" /></td> </tr></table>
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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.
  
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
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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 $| \zeta | > 1$ under a univalent conformal mapping (cf. also [[Conformal mapping|Conformal mapping]]) $z ( \zeta )$ of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014037.png" /></td> </tr></table>
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\begin{equation*} z ( \zeta ) = \zeta + \frac { a _ { 1 } } { \zeta } + \frac { a _ { 2 } } { \zeta ^ { 2 } } + \ldots \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014038.png" /></td> </tr></table>
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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 | > 1 , | \zeta ^ { \prime } | > 1. \end{equation*}
  
 
Then
 
Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014039.png" /></td> </tr></table>
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<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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014040.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014041.png" />. This gives also a procedure to evaluate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014042.png" /> numerically [[#References|[a2]]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014043.png" />, of course. There are many other such estimates in which the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014044.png" /> is involved [[#References|[a2]]]. As a simple consequence, there is the following very useful inequality for Jordan curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014045.png" /> with corners [[#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]]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014046.png" /></td> </tr></table>
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\begin{equation*} \frac { 1 } { \lambda } \geq | 1 - \alpha |. \end{equation*}
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014047.png" /> denotes the angle at the corner.
+
Here, $\alpha \pi$ denotes the angle at the corner.
  
For large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014048.png" /> one finds [[#References|[a2]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014049.png" /> must be contained in an annulus with radii <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014051.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014052.png" /> separates the boundary circles with these radii. However, in the other direction, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014053.png" /> can be close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014054.png" /> even though <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014055.png" /> lies, in the same manner, in an annulus for which the quotient of the radii is arbitrarily close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014056.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014057.png" /> [[#References|[a1]]], [[#References|[a2]]], [[#References|[a4]]]. If, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014058.png" /> is the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014059.png" /> under a univalent conformal mapping of an annulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014060.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014061.png" />), then
+
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 $r < | \zeta | < R$ ($r < 1 < R$), then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014062.png" /></td> </tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014064.png" />-quasi-conformal reflection at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014065.png" /> (i.e., a sense-reversing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014066.png" />-quasi-conformal mapping of the extended plane which leaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014067.png" /> pointwise fixed), then [[#References|[a2]]], [[#References|[a4]]]
+
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]]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014068.png" /></td> </tr></table>
+
\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]]]).
 
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]]]).
Line 53: Line 61:
 
From this Ahlfors inequality one obtains almost immediately [[#References|[a2]]], [[#References|[a4]]]:
 
From this Ahlfors inequality one obtains almost immediately [[#References|[a2]]], [[#References|[a4]]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014069.png" /></td> </tr></table>
+
\begin{equation*} \frac { 1 } { \lambda } \leq \operatorname { max } _ { \varphi } | \operatorname { cos } \alpha ( \varphi ) | \end{equation*}
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014070.png" /> is smooth and starlike with respect to the interior point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014071.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014072.png" /> denotes the angle between the ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014073.png" /> and the tangent at the point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014074.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120140/f12014075.png" />.
+
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]]].
 
For a theory of Fredholm eigenvalues for multiply-connected domains, see [[#References|[a3]]].
  
 
====References====
 
====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>
+
<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>

Latest revision as of 20:48, 23 January 2024

Let $C$ be a smooth 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 $> 1$ 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 and several integral equations to construct the conformal Riemann mapping function of $G$ [a1].

For arbitrary Jordan curves $C$, there is the following characterization of the Fredholm eigenvalue $\lambda$ [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).

One has: $1 \leq \lambda \leq \infty$, with $\lambda = \infty$ if and only if $C$ is a circle, and with $\lambda > 1$ if and only if $C$ is a quasi-circle [a4], [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).

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 $| \zeta | > 1$ under a univalent conformal mapping (cf. also 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 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 | > 1 , | \zeta ^ { \prime } | > 1. \end{equation*}

Then

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 [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 [a2]. As a simple consequence, there is the following very useful inequality for Jordan curves $C$ with corners [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 [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 $\lambda$ [a1], [a2], [a4]. If, for example, $C$ is the image of $| \zeta | = 1$ under a univalent conformal mapping of an annulus $r < | \zeta | < R$ ($r < 1 < R$), 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: 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 [a2], [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. [a2]).

From this Ahlfors inequality one obtains almost immediately [a2], [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 [a3].

References

[a1] D. Gaier, "Konstruktive Methoden der konformen Abbildung" , Springer (1964)
[a2] R. Kühnau, "Möglichst konforme Spiegelung an einer Jordankurve" Jahresber. Deutsch. Math. Ver. , 90 (1988) pp. 90–109
[a3] M. Schiffer, "Fredholm eigenvalues of multiply-connected domains" Pacific J. Math. , 9 (1959) pp. 211–269
[a4] G. Schober, "Estimates for Fredholm eigenvalues based on quasiconformal mapping" , Lecture Notes Math. , 333 , Springer (1973) pp. 211–217
How to Cite This Entry:
Fredholm eigenvalue of a Jordan curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fredholm_eigenvalue_of_a_Jordan_curve&oldid=17984
This article was adapted from an original article by Reiner Kühnau (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article