Teichmüller’s Modulsatz and the Variation of the Dirichlet Integral

Abstract

We show that changing the level curve of a harmonic function with the classical Hadamard variation with a small parameter entails a change in the Dirichlet integral of the function which is quadratic in the parameter. As a corollary, we supplement the well-known theorem of Teichmüller about the sum of moduli of doubly connected domains into which an annulus is subdivided by a continuum that differs little from a concentric circle.

1. Introduction

Consider the annulus $B=\{z:s<|z|<t\}$ with $0<s<t<\infty$ and denote by $\operatorname{mod}D$ the modulus of a doubly connected domain $D\subset{𝔺}$; in particular,

$$ \operatorname{mod}B=\frac{1}{2\pi}\log\frac{t}{s}. $$

Take some continuum $\gamma$ that separates $B$ into disjoint doubly connected domains $B_{1}$ and $B_{2}$. Grötzsch’s Lemma shows that

$$ \Delta(B,\gamma):=\operatorname{mod}B-\operatorname{mod}B_{1}-\operatorname{mod}B_{2} $$

(1)

is nonnegative and vanishes only in the case that $\gamma=\{z:|z|=r\}$ for an arbitrary $r$ with $s<r<t$. In 1938 Teichmüller established [1] the following: If

$$ \Delta(B,\gamma)\leq\delta $$

for $\delta>0$ sufficiently small then there is $C<\infty$, independent of $B$ and $\delta$, such that

$$ \frac{\sup\{|z|:z\in\gamma\}}{\inf\{|z|:z\in\gamma\}}\leq 1+C\sqrt{\delta\log\frac{1}{\delta}}. $$

This proposition is known in the literature as Teichmüller’s Modulsatz; see [2, Proposition 9.5; 3, Corollary 2.34; 4, Theorem 4.1], as well as [5, Chapter VI, Section 6, “narrow Modulsatz”]). Teichmüller pointed out [1] the accuracy of his estimate understood in the sense that for every $\varepsilon>0$ there exists a continuum $\gamma_{\varepsilon}$ avoiding some concentric circle in $B$ by $\varepsilon$, while

$$ \Delta(B,\gamma_{\varepsilon})\leq\delta(\varepsilon),\quad\delta(\varepsilon)\asymp\frac{\varepsilon^{2}}{\log\frac{1}{\varepsilon}}\quad\text{as }\varepsilon\to 0. $$

Bertilsson gave [3, Example 2.26] an explicit form of such continuum in the “dual problem”; see also [4, Chapter V, Exercise 10]. The factor $1/\bigl{(}\log\frac{1}{\varepsilon}\bigr{)}$ appears because one point of the continuum $\gamma(\varepsilon)$ approaches the circle as $\varepsilon\to 0$ one order in $\varepsilon$ slower than the others. It is natural to suppose that a more uniform convergence of $\gamma_{\varepsilon}$ to the circle would ensure that

$$ \Delta(B,\gamma_{\varepsilon})=O(\varepsilon^{2})\quad\text{as }\varepsilon\to 0. $$

(2)

Indeed, [6] observes that (2) holds whenever we obtain $\gamma_{\varepsilon}$ from a concentric circle via some Hadamard deformation, defined in general as follows. Take a smooth curve $\gamma$ in ${𝔺}$ and a real twice continuously differentiable function $\varphi$ on $\gamma$. Given a sufficiently small $\varepsilon>0$, define the “deformation” of $\gamma$ as

$$ \delta n(z):=\varepsilon\varphi(z)+O(\varepsilon^{2}) $$

(3)

such that $\gamma$ goes into the curve $\gamma_{\varepsilon}=\{z_{\varepsilon}=z+\delta n(z)i\,dz/|dz|:z\in\gamma\}$. Here $\delta n(z)$ is a twice continuously differentiable function on $\gamma$ and $O(\varepsilon^{2})$ admits on $\gamma$ a uniform estimate.¹ In the case (3) we can prove (2) using Hadamard’s variational formula for the Dirichlet integral [7, (A3.11); 6, (2.2)]. In [6] we used (2) substantially to obtain a fine property of the Green’s energy of a discrete charge.

In this note we give a direct proof of a more general result than (2). Moreover, we pass from doubly connected domains to arbitrary ones. In this regard, instead of comparing the moduli of annular domains, we study the behavior of the Dirichlet integral of a harmonic of function when its level curves changes via the deformation in (3). Now we proceed to precise statements.

Given a finite domain $B$ in the plane ${𝔺}$ whose boundary consists of analytic arcs and closed analytic Jordan curves, consider a nonconstant function $u$ continuous on $\overline{B}$, harmonic on $B$, and satisfying the boundary conditions of the mixed Dirichlet problem [4, Theorem B.4]. More exactly, on some closed arcs (curves) $\Gamma_{1}$ of the boundary of $B$ it takes constant values, while on the remaining parts $\Gamma_{2}$ of the boundary of $B$ the normal derivative $\partial u/\partial n$ of $u$ vanishes; the latter set can be empty. Consider some collection $\{\gamma\}$ consisting of finitely many disjoint closed Jordan arcs or closed Jordan curves in $B$ lying on (possibly distinct) level curves of $u$². To each curve $\gamma\in\{\gamma\}$ associate the curve $\gamma_{\varepsilon}$ with $\varepsilon>0$ obtained from $\gamma$ via the deformation in (3), where $\varphi$ is a real twice continuously differentiable function defined on the union $\bigcup\gamma$, while $\varphi\not\equiv 0$ on $\bigcup\gamma$ and the support of $\delta n(z)$ avoids the endpoints of $\gamma\in\{\gamma\}$. Henceforth $\cup$ and $\Sigma$ stand for the union and the sum over all curves $\gamma\in\{\gamma\}$. Assume that $\varepsilon$ is so small that all curves $\gamma_{\varepsilon}$ are pairwise disjoint and lie in $B$. Suppose that the function $u_{\varepsilon}$ is continuous on $\overline{B}$, harmonic on $B_{\varepsilon}:=B\setminus\bigcup\gamma_{\varepsilon}$, satisfies the boundary conditions for $u$ on $\partial B$, and on each curve $\gamma_{\varepsilon}$ takes the constant value equal to the value of $u$ on the curve $\gamma$ corresponding under the deformation in (3). Put

$$ I(v,\Omega)=\iint\limits_{\Omega}|\nabla v|^{2}\,dxdy. $$

Theorem 1

Under the above conditions we have the asymptotic equality

$$ I(u_{\varepsilon},B_{\varepsilon})-I(u,B)\asymp\varepsilon^{2}\quad\text{as }\varepsilon\to 0. $$

(4)

Observe that the left-hand side in (4) is nonnegative by the Dirichlet principle.

The proof of (4) rests substantially on Kellogg’s results about the behavior of partial derivatives of a harmonic function on the boundary of its domain of definition [8].

We confine ourselves to the case that $u$ and $u_{\varepsilon}$ are potential functions for generalized condensers [9]. It is clear from the proof of Theorem 1 that (4) also holds if we replace the boundary conditions for these functions by the existence and continuity of their first partial derivatives in a neighborhood of $\partial B$.

In connection with (2) and (4), the assumption comes up that

$$ \Delta(B,\gamma_{\varepsilon})\asymp\varepsilon^{2}\quad\text{as }\varepsilon\to 0 $$

(5)

is valid. However, $\Delta(B,\gamma_{\varepsilon})=0$ for $\delta n(z)\equiv c\varepsilon$, where $c$ is a constant. The author is aware of examples of concrete deformations (3) for which (5) indeed holds. Possibly, $\delta n(z)\equiv c\varepsilon$ is the unique case for which this fails.

The final part of this article gives a corollary to Theorem 1 in the case that $B$ is a circular annulus, see the inequality in (9). We show that this corollary also yields (2).

2. Proof of Theorem 1

We may assume that the boundary of $B$ consists of analytic Jordan curves. Consider the function

$$ f_{\varepsilon}=\frac{u-u_{\varepsilon}}{\varepsilon} $$

on $B_{\varepsilon}$ and some function $f$ which is harmonic on $B\setminus\bigcup\gamma$, continuous on $\overline{B}$, and satisfies the boundary conditions

$$ f=0\text{ on }\Gamma_{1},\quad\frac{\partial f}{\partial n}=0\text{ on }\Gamma_{2},\quad f(z)=\varphi(z)\frac{\partial u}{\partial n}(z),\ z\in\gamma\ \forall\gamma\in\{\gamma\}. $$

Henceforth, differentiation is with respect to the positively oriented normal to the corresponding curve. In view of the uniform continuity of $f$, for every real $\delta>0$ and arbitrary curve $\gamma\in\{\gamma\}$ we have

$$ |f(z)-f(z_{\varepsilon})|<\delta,\quad z\in\gamma, $$

for $\varepsilon$ sufficiently small. Taylor’s formula yields

$$ f_{\varepsilon}(z_{\varepsilon})=\varphi(z)\frac{\partial u}{\partial n}(z)+O(\varepsilon)=f(z)+O(\varepsilon)\quad\text{as }\varepsilon\to 0; $$

furthermore, $O(\varepsilon)$ is uniform in $z\in\gamma$. Hence, $|f_{\varepsilon}(z_{\varepsilon})-f(z_{\varepsilon})|\leq\delta$ for all $z_{\varepsilon}\in\gamma_{\varepsilon}$ and $\varepsilon$ sufficiently small. The maximum principle for harmonic functions and Hopf’s Lemma imply that $|f_{\varepsilon}(z)-f(z)|\leq\delta$ for all $z\in B_{\varepsilon}$, and consequently, on an arbitrary compact subset of $B\setminus\bigcup\gamma$ for $\varepsilon$ small. Thus, $f_{\varepsilon}$ together with partial derivatives converge to $f$ as $\varepsilon\to 0$ uniformly inside $B\setminus\bigcup\gamma$.

Associate to each arc $\gamma_{\varepsilon}$ with $\gamma\in\{\gamma\}$ a doubly connected domain $Q_{\gamma_{\varepsilon}}$ with one boundary component $\gamma_{\varepsilon}$ and the other some closed analytic Jordan curve. Associate to the closed curve $\gamma_{\varepsilon}$ with $\gamma\in\{\gamma\}$ two disjoint doubly connected domains $Q^{+}_{\gamma_{\varepsilon}}$ and $Q^{-}_{\gamma_{\varepsilon}}$ with one boundary component $\gamma_{\varepsilon}$ and the other a closed analytic Jordan curve. Assume that $Q_{\gamma_{\varepsilon}}$, $Q^{+}_{\gamma_{\varepsilon}}$, and $Q^{-}_{\gamma_{\varepsilon}}$ are disjoint and the closures of the domains lie in $B$. By Kellogg’s Theorem [8, Theorem 1] we conclude that $f_{\varepsilon}$ has continuous first partial derivatives on the closures of $Q_{\gamma_{\varepsilon}}$, $Q^{+}_{\gamma_{\varepsilon}}$, and $Q^{-}_{\gamma_{\varepsilon}}$. We will need the normal derivative $\partial f_{\varepsilon}/\partial n$ on the boundaries of the domains to be bounded uniformly in $\varepsilon$.

The boundary value problem for harmonic functions, in our case for the function $f_{\varepsilon}$, is reduced in [8] to integral equations. We can express the solution $f_{\varepsilon}$, in $Q^{+}_{\gamma_{\varepsilon}}$ for definiteness, as the sum of a double layer potential $W_{\varepsilon}$ and a single layer potential $V_{\varepsilon}$ [8, Section 3]. The partial derivatives of the potentials on the boundary of $Q^{+}_{\gamma_{\varepsilon}}$ are integrals over the boundary of $Q^{+}_{\gamma_{\varepsilon}}$ of some functions depending continuously on the domain $Q^{+}_{\gamma_{\varepsilon}}$ (with respect to $\varepsilon$), as well as, in the case of $W_{\varepsilon}$, on the derivative $\partial f_{\varepsilon}/\partial s$ of the boundary value of $f_{\varepsilon}$ along the tangent to the boundary of $Q^{+}_{\gamma_{\varepsilon}}$; see pp. 111, 114, 120, and Section 6 in [8]. It is clear from the above that $\partial f_{\varepsilon}/\partial s$ is continuous and bounded uniformly in $\varepsilon$ both on the curve $\gamma_{\varepsilon}$ by definition and on $(\partial Q^{+}_{\gamma_{\varepsilon}})\setminus\gamma_{\varepsilon}$ by the uniform convergence on $B\setminus\bigcup\gamma$ of the partial derivative of $f_{\varepsilon}$. In view of the expression (22) of [8], we conclude that the first partial derivatives of $f_{\varepsilon}$ are bounded on $\partial Q^{+}_{\gamma_{\varepsilon}}$ uniformly in $\varepsilon$. We verify similarly that the first partial derivatives of $f_{\varepsilon}$ on $\partial Q^{-}_{\gamma_{\varepsilon}}$ are bounded uniformly in $\varepsilon$. In the case of $Q_{\gamma_{\varepsilon}}$ these derivatives are also bounded, which we can easily verify by mapping $Q_{\gamma_{\varepsilon}}$ conformally onto a Jordan domain and applying previous arguments to the corresponding superposition; the support of $\delta n(z)$ does not contain the endpoints of $\gamma$.

Henceforth we denote the curve $\gamma\in\{\gamma\}$ also by $\gamma^{+}$, whereas the same curve with the opposite direction by $\gamma^{-}$. Similarly, $\gamma^{+}_{\varepsilon}=\gamma_{\varepsilon}$ and $\gamma^{-}_{\varepsilon}$ is the curve opposite to $\gamma^{+}_{\varepsilon}$. Applying the Green’s formula,³ we obtain

$$ \begin{gathered}\displaystyle I(u_{\varepsilon},B_{\varepsilon})-I(u,B)=I(u_{\varepsilon},B_{\varepsilon})-I(u,B_{\varepsilon})\\ \displaystyle-2\sum\biggl{[}\,\int\limits_{\gamma^{+}_{\varepsilon}}(u-u_{\varepsilon})\frac{\partial u}{\partial n}\,|dz|+\int\limits_{\gamma^{-}_{\varepsilon}}(u-u_{\varepsilon})\frac{\partial u}{\partial n}\,|dz|\biggr{]}\\ \displaystyle=I(u_{\varepsilon},B_{\varepsilon})+I(u,B_{\varepsilon})+2\int\limits_{\partial B_{\varepsilon}}(u_{\varepsilon}-u+u)\frac{\partial u}{\partial n}\,|dz|=I(u-u_{\varepsilon},B_{\varepsilon})\\ \displaystyle=-\int\limits_{\partial B_{\varepsilon}}(u-u_{\varepsilon})\frac{\partial(u-u_{\varepsilon})}{\partial n}\,|dz|\\ \displaystyle\text{(the boundary of $B$ is oriented in the positive direction)}\\ \displaystyle=-\sum\biggl{[}\int\limits_{\gamma^{+}_{\varepsilon}}(u-u_{\varepsilon})\frac{\partial(u-u_{\varepsilon})}{\partial n}\,|dz|+\int\limits_{\gamma^{-}_{\varepsilon}}(u-u_{\varepsilon})\frac{\partial(u-u_{\varepsilon})}{\partial n}\,|dz|\biggr{]}.\end{gathered} $$

Among the above relations we highlight the two equalities

$$ I(u_{\varepsilon},B_{\varepsilon})-I(u,B)=-\sum\biggl{[}\,\int\limits_{\gamma^{+}_{\varepsilon}}(u-u_{\varepsilon})\frac{\partial(u-u_{\varepsilon})}{\partial n}\,|dz|+\int\limits_{\gamma^{-}_{\varepsilon}}(u-u_{\varepsilon})\frac{\partial(u-u_{\varepsilon})}{\partial n}\,|dz|\biggr{]}, $$

(6)

$$ I(u_{\varepsilon},B_{\varepsilon})-I(u,B)=I(u-u_{\varepsilon},B_{\varepsilon}). $$

(7)

Appreciating the above information about $f_{\varepsilon}$, we arrive at the estimate

$$ \biggl{|}\int\limits_{\gamma^{\pm}_{\varepsilon}}(u-u_{\varepsilon})\frac{\partial(u-u_{\varepsilon})}{\partial n}\,|dz|\biggr{|}=\varepsilon^{2}\biggl{|}\int\limits_{\gamma^{\pm}_{\varepsilon}}f_{\varepsilon}\frac{\partial f_{\varepsilon}}{\partial n}\,|dz|\biggr{|}=O(\varepsilon^{2})\quad\text{as }\varepsilon\to 0. $$

Consequently, (6) yields

$$ I(u_{\varepsilon},B_{\varepsilon})-I(u,B)=O(\varepsilon^{2})\quad\text{as }\varepsilon\to 0. $$

To prove the inverse relation with (6) would require a sharper estimate for the derivative $\partial f_{\varepsilon}/\partial n$, and consequently a deeper analysis of the proof of Theorem 1 in [8]. It is simpler to observe that under the hypotheses of Theorem 1 there is a subarc $\gamma_{0}\subset\gamma$, $\gamma\in\{\gamma\}$ on which $\varphi(z)\neq 0$, while Hopf’s Lemma yields $\partial u/\partial n\neq 0$ on $\gamma_{0}$. Consequently, $f\not\equiv 0$ in $B\setminus\bigcup\gamma$. Thus, $B\setminus\bigcup\gamma$ includes a closed disk $E$ with $I(f,E)\neq 0$. For $\varepsilon$ sufficiently small the disk $E$ lies in $B_{\varepsilon}$ and

$$ I(u-u_{\varepsilon},B_{\varepsilon})\geq I(u-u_{\varepsilon},E)=\varepsilon^{2}I(f_{\varepsilon},E)=\varepsilon^{2}I(f,E)+o(\varepsilon^{2})\geq\varepsilon^{2}\frac{I(f,E)}{2}. $$

With (7) this yields

$$ \varepsilon^{2}=O(I(u_{\varepsilon},B_{\varepsilon})-I(u,B))\quad\text{as }\varepsilon\to 0. $$

The proof of Theorem 1 is complete.

3. Moduli and Capacities

To an arbitrary doubly connected domain $D\subset{𝔺}$ with nondegenerate boundary components $E_{0}$ and $E_{1}$, associate the condenser $C=(E_{0},E_{1})$ whose capacity equals $\operatorname{cap}C=I(\omega,D)$. Here $\omega$ is the “potential function” of $C$ continuous on $\overline{D}$, harmonic on $D$, vanishing on $E_{0}$, and equal to 1 on $E_{1}$. It is well known that

$$ \operatorname{cap}C=\frac{1}{\operatorname{mod}D}; $$

for more detail on capacity, see [9]. Put

$$ B(\tau_{1},\tau_{2})=\{z:\tau_{1}<|z|<\tau_{2}\},\quad T(\tau)=\{z:|z|=\tau\}. $$

The circle $\gamma=T(r)$ is a level curve of the function

$$ u(z)=\frac{\log(|z|/t)}{\log(s/t)},\quad 0<s<r<t<\infty. $$

The curve $\gamma_{\varepsilon}$ resulting from $\gamma$ by the deformation in (3) partitions the annulus $B=B(s,t)$ into disjoint doubly connected domains $B_{1}$ and $B_{2}$; assume that $T(s)\subset\partial B_{1}$. In the case $\varphi\not\equiv 0$ Theorem 1 yields

$$ I(u_{\varepsilon},B_{1}\cup B_{2})-I(u,B)\asymp\varepsilon^{2}\quad\text{as }\varepsilon\to 0, $$

(8)

where $u_{\varepsilon}$ is a function continuous on $\overline{B}$, harmonic on $B_{1}\cup B_{2}$, equal to 1 on $T(s)$ and to 0 on $T(t)$, and

$$ u_{\varepsilon}=\delta:=\frac{\log(r/t)}{\log(s/t)}\text{ on }\gamma_{\varepsilon}. $$

The function $u$ is potential for the condenser $C=(T(t),T(s))$, while $(u_{\varepsilon}-\delta)/(1-\delta)$ is the potential function for the condenser $C_{1}=(\gamma_{\varepsilon},T(s))$ and $u_{\varepsilon}/\delta$ is the potential function for the condenser $C_{2}=(T(t),\gamma_{\varepsilon})$. Thus, (8) becomes

$$ 0\leq(1-\delta)^{2}\operatorname{cap}C_{1}+\delta^{2}\operatorname{cap}C_{2}-\operatorname{cap}C\asymp\varepsilon^{2}\quad\text{as }\varepsilon\to 0. $$

In terms of moduli this inequality looks like

$$ 0\leq\frac{\operatorname{mod^{2}}B^{*}_{1}}{\operatorname{mod}B_{1}}+\frac{\operatorname{mod^{2}}B^{*}_{2}}{\operatorname{mod}B_{2}}-\operatorname{mod}B\asymp\varepsilon^{2}\quad\text{as }\varepsilon\to 0, $$

(9)

where $B^{*}_{1}=B(s,r)$ and $B^{*}_{2}=B(r,t)$, while the deformation of (3) satisfies the condition $\varphi\not\equiv 0$.

Verify that (9) implies (2). We may assume that $\varphi\not\equiv 0$. Denote by $B^{\prime}_{1}$ and $B^{\prime}_{2}$ the circular annuli $B(s,r(\varepsilon))$ and $B(r(\varepsilon),t)$ whose areas in the logarithmic metric $(2\pi|z|)^{-1}|dz|$ are equal respectively to the areas of the domains $B_{1}$ and $B_{2}$ in the same metric. It is obvious that $r(\varepsilon)=r+c\varepsilon+O(\varepsilon^{2})$, where $c$ is some constant depending on the function $\varphi$. Rengel’s Lemma [9, Section 5.5] yields

$$ \operatorname{mod}B_{1}\leq\operatorname{mod}B^{\prime}_{1},\quad\operatorname{mod}B_{2}\leq\operatorname{mod}B^{\prime}_{2}. $$

Subtracting from (9) the equality

$$ \frac{\operatorname{mod^{2}}B^{*}_{1}}{\operatorname{mod}B^{\prime}_{1}}+\frac{\operatorname{mod^{2}}B^{*}_{2}}{\operatorname{mod}B^{\prime}_{2}}-\operatorname{mod}B=O(\varepsilon^{2})\quad\text{as }\varepsilon\to 0, $$

we obtain

$$ \frac{\operatorname{mod^{2}}B^{*}_{1}}{\operatorname{mod}B_{1}}-\frac{\operatorname{mod^{2}}B^{*}_{1}}{\operatorname{mod}B^{\prime}_{1}}+\frac{\operatorname{mod^{2}}B^{*}_{2}}{\operatorname{mod}B_{2}}-\frac{\operatorname{mod^{2}}B^{*}_{2}}{\operatorname{mod}B^{\prime}_{2}}\leq C\varepsilon^{2}, $$

where $C$ is some constant. Hence,

$$ \operatorname{mod}B^{\prime}_{k}-\operatorname{mod}B_{k}=O(\varepsilon^{2})\quad\text{as }\varepsilon\to 0,\ k=1,2. $$

Consequently,

$$ \begin{gathered}\displaystyle 0\leq\operatorname{mod}B-\operatorname{mod}B_{1}-\operatorname{mod}B_{2}=\operatorname{mod}B^{\prime}_{1}-\operatorname{mod}B_{1}+\operatorname{mod}B^{\prime}_{2}-\operatorname{mod}B_{2}=O(\varepsilon^{2})\quad\text{as }\varepsilon\to 0,\end{gathered} $$

which means the validity of (2). Similarly we can establish the inequalities supplementing Theorem 4.2 and Corollary 4.3 of [4].