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Variational formulae for conformal capacity V. N. Dubinin Institute for Applied Mathematics, Far Eastern Branch, Russian Academy of Sciences, Vladivostok, Russia
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     § 1. Introduction and statements of the main resultsConsider the simplest condenser $C=\{B,\gamma,(\partial B)\,{\setminus}\, \gamma\}$ in the complex plane $\mathbb{C}$. Here $B$ is a subdomain of $\mathbb{C}$, bounded by a finite system of $C^2$-curves and $\gamma$ is a boundary component of this domain such that $(\partial B)\,{\setminus}\, \gamma\neq\varnothing$. Then $B$ is called the field of the condenser $C$, and the sets $\gamma$ and $(\partial B) \setminus \gamma$ are its plates. The conformal capacity $\operatorname{cap} C$ of $C$ is defined to be the greatest lower bound of the Dirichlet integrals over all real functions $v$ that are continuous on $\overline{B}$, satisfy the Lipschitz condition locally in $B$ and are equal to 1 on $\gamma$ and to 0 on $(\partial B)\setminus \gamma$. By Dirichlet’s principle the capacity of $C$ is equal to the Dirichlet integral of the potential function of this condenser, which is harmonic in $B$, continuous in $\overline{B}$, equal to $1$ on $\gamma$ and to $0$ on $(\partial B)\setminus \gamma$ (see [1]). Questions of the behaviour of various types of condenser capacity under deformations of the field $B$ were considered by many authors, who were motivated by numerous applications to geometric function theory, potential theory and related areas (for instance, see [1]–[10] and the bibliography there). Note, in particular, Nasyrov’s paper [9], in which, using the techniques of conformal mappings and extremal metrics, the author investigated the variations of the moduli of quadrilaterals and doubly connected domains, reduced moduli and Robin capacities under sufficiently smooth changes of the boundaries of the domains in question. Let $z=z(s)$ be the equation of the curve $\gamma$ endowed with the positive orientation relative to the domain (where $s$ is a natural parameter), and let $\varphi$ be a real, twice continuously differentiable function on $\gamma$. Following Hadamard, for small $\varepsilon>0$ we define a ‘deformation’ of $\gamma$ which takes $\gamma$ to the curve $\gamma^*$ defined by the equation Let $B^*$ denote the domain with boundary formed by $(\partial B)\setminus \gamma$ and the curve $\gamma^*$. Then for the Green’s functions of $B$ and $B^*$ we have the classical Hadamard formula (see [1], (A3.3))
$$
\begin{equation*}
g_{B^*}(\zeta,\eta)=g_{B}(\zeta,\eta)-\frac{\varepsilon}{2\pi}\int_\gamma \frac{\partial g_B(z,\zeta)}{\partial n_z} \,\frac{\partial g_B(z,\eta)}{\partial n_z}\, \varphi(s)\,ds+O(\varepsilon^2)
\end{equation*}
\notag
$$
(also see [11] and the bibliography there). Here and throughout, $\partial/\partial n$ denotes differentiation along the inward normal, and asymptotic equalities are considered as $\varepsilon\to 0$. From this, differentiating twice under the integral sign, we can easily obtain an asymptotic formula for the harmonic measure of $\gamma$ and then the following formula for the variation of the conformal capacity (see [1], (A3.11)):
$$
\begin{equation}
\operatorname{cap} C^*=\operatorname{cap} C+\varepsilon\int_{\gamma}\biggl(\frac{\partial u}{\partial n}\biggr)^2\varphi(s)\,ds+O(\varepsilon^2),
\end{equation}
\tag{1.2}
$$
where $C^*=\{B^*,\gamma^*,(\partial B)\setminus \gamma\}$ and $u$ is the potential function of the condenser $C$. From (1.2) we obtain a variational formula for the capacity of a condenser with many plates (see [10]), provided that the boundary of the field of the condenser lies in the union of these plates (see [1], (A3.12)). In this paper we present a direct proof of variational formulae for the conformal capacity treated in a broader sense, when both the boundary equipotential lines ($u=\mathrm{const}$) and the boundary streamlines (if any) are deformed. Let us turn to precise statements. Let $B$ be an open set in the complex plane $\overline{\mathbb{C}}$. By a generalized condenser in $\overline{B}$ we mean a triple $C=(B,\mathcal{E},\Delta)$, where $\mathcal{E}=\{E_k\}^n_{k=1}$ is a system of closed nonempty pairwise disjoint sets $E_k\subset \overline{B}$, $k=1,\dots,n$, and $\Delta=\{\delta_k\}_{k=1}^n$ is a set of real numbers $\delta_k$, $k=1,\dots,n$, $n\geqslant 2$. We call the set $\overline{B}\setminus \bigcup_{k=1}^nE_k$, which is open in $\overline{B}$, the field of the condenser $C$, and we call $E_k$ plates of this condenser and $\delta_k$ the levels of the potential or, briefly, potentials of the plates $E_k$, $k=1,\dots,n$. The capacity $\operatorname{cap}C$ of $C$ is by definition the greatest lower bound of the Dirichlet integrals ${I(v,B\setminus \bigcup_{k=1}^n E_k)}$ over all real-valued functions $v$ that are continuous in $\overline{B}$, satisfy the Lipschitz condition on compact subsets of $B$ and are equal to $\delta_k$ on $E_k$, $k=1,\dots,n$. If the set $(\partial B)\setminus \bigcup_{k=1}^n E_k$ is either empty or consists of a finite number of analytic curves and there exists a continuous function $u$ in $\overline{B}$ that is harmonic in $\overline{B}\setminus \bigcup_{k=1}^n E_k$, equal to $\delta_k$ on $E_k$, $k=1,\dots,n$, and satisfies $\partial u/\partial n=0$ on ${(\partial B)\setminus \bigcup_{k=1}^n E_k}$, then we call it the potential function of the condenser $C$. Using conformal mappings, these definitions can be extended to a wider class of condensers. In what follows, without loss of generality we consider only condensers $C=(B,\mathcal{E},\Delta)$ such that $B$ is a domain in the plane $\mathbb{C}$ bounded by a finite number of Jordan curves, and plates in the system $\mathcal{E}$ consist of a finite number of nondegenerate connected components lying on the boundary of $B$. Such a condenser possesses a potential function $u$, and we have (see [10], § A1). By the curve $\gamma$ we mean the union of a finite number of analytic arcs or closed analytic curves in $\mathbb{C}$ which lie on the boundary of $B;$ $\gamma^*$ is the union of the curves obtained from $\gamma$ by a deformation of the form (1.1) such that the support of the function $\varphi(s)$ does not contain endpoints of $\gamma$ and, finally, $B^*$ is the domain bounded by the curves $((\partial B)\setminus\gamma)\cup \gamma^*$. Then the following results hold.Theorem 1. Let the condenser $C=(B,\{E_k\}_{k=1}^n,\{\delta_k\}_{k=1}^n)$ and the curve $\gamma$ be as defined above, and let $\gamma$ lie in the union of the plates of $C$. Then equality (1.2) holds, where and $u$ is the potential function of the condenser $C$. Theorem 2. If the curve $\gamma$ is disjoint from the plates of the condenser $C=(B,\{E_k\}_{k=1}^n, \{\delta_k\}_{k=1}^n)$, then Formulae (1.2) and (1.3) refine quantitatively some variational principles for conformal mappings (for instance, see [2], [3] and [12]). It is also known that the Dirichlet integral has various physical interpretations. In particular, (1.2) and (1.3) can be regarded as variational formulae for the energy of an electrostatic field (see [1], § 1.1, or [4]). We present the proofs of Theorems 1 and 2 in §§ 2 and 3, respectively. We also comment there on our results. In § 4 we consider variational formulae for quadratic forms with coefficients depending on inner radii, Robin radii, and Green’s or Robin functions. § 2. Deformation of equipotential lines Proof of Theorem 1. Bearing in mind the conformal invariance of the Dirichlet integral we can assume that the boundary of $B$ consists of a finite number of analytic Jordan curves. Let $u$ be the potential function of $C$ and $u^*$ be the potential function of $C^*$. We can extend $u$ to a harmonic function in a neighbourhood of $\gamma$. We denote this extension by the same symbol $u$ and assume that $\varepsilon$ is sufficiently small so that $\gamma^*$ lies in the neighbourhood just mentioned. Thus, the function $u-u^*$ is harmonic in the domain $B^*$, continuous in its closure, vanishes on $\bigcup_{k=1}^n E_k\setminus\gamma$, and its normal derivative at the points in $(\partial B)\setminus\bigcup_{k=1}^n E_k$ is zero. On $\gamma^*$ we have
$$
\begin{equation*}
u(z)-u^*(z)=\varepsilon\varphi(s)\,\frac{\partial u}{\partial n}(z(s))+O(\varepsilon^2).
\end{equation*}
\notag
$$
Here the variables $z$ and $s$ are related by an equality $z=z^*(s)$. Taking Hopf’s lemma into account, from the maximum principle for harmonic functions we conclude that uniformly in $z\in B^*$ and $\varepsilon>0$. It follows from Kellogg’s results in [13]1 that the first partial derivatives of the function $(u-u^*)/\varepsilon$ are continuous and bounded (uniformly in $\varepsilon$) in the part of a closed neighbourhood of $\gamma^*$ that lies in $\overline{B}^*$. Hence, using Green’s formula2 we obtain the estimate
$$
\begin{equation}
\begin{aligned} \, I(u-u^*,B^*) &=-\int_{\partial B^*}(u-u^*)\,\frac{\partial (u-u^*)}{\partial n}\,ds \nonumber \\ &=-\int_{\gamma^*}(u-u^*)\,\frac{\partial (u-u^*)}{\partial n}\,ds=O(\varepsilon^2). \end{aligned}
\end{equation}
\tag{2.1}
$$
Applying Green’s formula again we obtain
$$
\begin{equation*}
\begin{aligned} \, I(u-u^*,B^*) &=I(u,B^*)+I(u^*,B^*)+2\int_{\partial B^*}(u^*-u+u)\,\frac{\partial u}{\partial n}\,ds \\ &=I(u^*,B^*)-I(u,B^*)-2\int_{\gamma^*}(u-u^*)\,\frac{\partial u}{\partial n}\,ds. \end{aligned}
\end{equation*}
\notag
$$
Note that
$$
\begin{equation*}
I(u,B^*)+\int_{\gamma^*}u\,\frac{\partial u}{\partial n}\,ds= I(u,B)+\int_{ \gamma}u\,\frac{\partial u}{\partial n}\,ds,
\end{equation*}
\notag
$$
and the function $u$ takes a constant value on each component of $\gamma$ which is equal to the value of $u^*$ on the corresponding component of $\gamma^*$. Therefore,
$$
\begin{equation*}
\begin{aligned} \, &I(u-u^*,B^*) \\ &\qquad=I(u^*,B^*)-I(u,B)+\int_{\gamma^*}u\,\frac{\partial u}{\partial n}\,ds -\int_{\gamma}u\,\frac{\partial u}{\partial n}\,ds -2\int_{\gamma^*}(u-u^*)\,\frac{\partial u}{\partial n}\,ds \\ &\qquad=I(u^*,B^*)-I(u,B)-\int_{\gamma}u\,\frac{\partial u}{\partial n}\,ds +\int_{\gamma^*}u^*\,\frac{\partial u}{\partial n}\,ds -\int_{\gamma^*}(u-u^*)\,\frac{\partial u}{\partial n}\,ds \\ &\qquad=I(u^*,B^*)-I(u,B) \\ &\qquad\qquad-\int_{\gamma}\biggl[\varepsilon \varphi(s)\,\frac{\partial u}{\partial n}(z(s))+O(\varepsilon^2)\biggr] \biggl[\frac{\partial u}{\partial n}(z(s))\,{+}\,O(\varepsilon)\biggr]ds+O(\varepsilon^2) \\ &\qquad=I(u^*,B^*)-I(u,B)-\varepsilon\int_{\gamma} \biggl(\frac{\partial u}{\partial n}\biggr)^2 \varphi(s)\,ds+O(\varepsilon^2). \end{aligned}
\end{equation*}
\notag
$$
Taking (2.1) into account, this yields (1.2).
Theorem 1 is proved. To keep the matter simple we have limited ourselves to deformations (1.1) of the classical form, with a $C^2$-function $\varphi$ (see [1], § 3.1). Formula (1.2) holds under the assumption that $\varphi(s)$ is a piecewise smooth function of the arc length $s$. Furthermore, it is clear from the proof of Theorem 1 that the deformation (1.1) can be replaced by a transition from $\gamma$ to an arbitrary smooth curve $\gamma^*$ with the same endpoints as $\gamma$ and a well-defined projection onto $\gamma$. For example, if $\gamma^*\subset\overline{B}$, then formula (1.2) also holds once we replace the function $\varphi(s)$ by the linear measure of the intersection of $B\setminus \overline{B}^*$ with the normal to $\gamma$ at $z(s)$. Another obvious supplement to (1.2) following from the proof of Theorem 1 is briefly as follows. Assume that the function $\varphi$ in the deformation (1.1) is defined on a compact subset of $\gamma$ depending on $\varepsilon$ and of linear measure $O(\varepsilon)$. Then we also have (1.2), and we can write $O(\varepsilon^3)$ in place of $O(\varepsilon^2)$. The condition that $\gamma$ is analytic can also be relaxed by considering $C^2$-curves. In this case we must prove Theorem 1 for $\varphi\geqslant0$ first and then consider the general case similarly to [1], § 3.1. For an explicit example of formula (1.2) consider the quadrilateral ${B=\{z=x+i y\colon|x|<1,\,0<y<1\}}$, the arc $\gamma=[-1,1]$ and the condenser ${C=(B,\{\gamma,[-1+ i,1+i]\},\{0,1\})}$. Then the potential function is $u(x+iy)=y$. From the asymptotic equality (1.2) we obtain
$$
\begin{equation*}
\operatorname{cap}C^*=2+\varepsilon\int_{\gamma}\varphi(s)\,ds+O(\varepsilon^2) =2+\kappa+O(\varepsilon^2),
\end{equation*}
\notag
$$
where $\kappa$ is the total area of the curvilinear trapezoid bounded by $\gamma^*$ and the $x$-axis, which is taken with plus sign when the corresponding part of the trapezium lies above the real axis and with minus sign otherwise (cf. [9], Corollary 2). Note that terms in (1.2) are conformal invariants. This can be useful in calculations of particular variations. For example, let $B$ be the upper half-plane, let $\gamma=[-1,1]$, let $k$ be a number satisfying $0<k<1$, and assume that the potential $u$ is zero on $[-1,1]$ and one on $[-\infty,-1/k]\cup[1/k,+\infty]$. Then the function maps the domain $B$ conformally onto the quadrilateral $\{w=\xi+i\eta\colon |\xi|<\mathbf K(k), {0<\eta<\mathbf K'(k)}\}$, where $\mathbf K(k)$ and $\mathbf K'(k)$ are complete elliptic integrals of the first kind (for instance, see [2], § 39). Therefore,
$$
\begin{equation*}
\begin{gathered} \, u(z)=\frac{\operatorname{Im}f(z)}{\mathbf K'(k)}, \qquad \operatorname{cap}C=\frac{2\mathbf K(k)}{\mathbf K'(k)}, \\ \frac{\partial u}{\partial n}=\frac{\partial u}{\partial y}=\frac{1}{\mathbf K'(k)}f'(x)=\frac{1}{\mathbf K'(k)\sqrt{(1-x^2)(1-k^2x^2)}}, \qquad x\in (-1,1), \end{gathered}
\end{equation*}
\notag
$$
and formula (1.2) yields
$$
\begin{equation*}
\operatorname{cap}C^*=\frac{2\mathbf K(k)}{\mathbf K'(k)}+\frac{\varepsilon}{(\mathbf K'(k))^2}\int_{-1}^1\frac{\varphi(x)\,dx}{(1-x^2)(1-k^2x^2)}+O(\varepsilon^2).
\end{equation*}
\notag
$$
§ 3. Deformation of streamlines Proof of Theorem 2. Let $B$ and the potential functions $u$ and $u^*$ be as in the proof of Theorem 1. Following this proof, consider the Dirichlet integral
$$
\begin{equation*}
\begin{aligned} \, I(u-u^*,B^*) &=I(u^*,B^*)-I(u,B^*)-2\int_{\gamma^*}(u-u^*)\,\frac{\partial u}{\partial n}\,ds \\ &=I(u^*,B^*)-I(u,B)+2\int_{\gamma^*}u^*\,\frac{\partial u}{\partial n}\,ds -\int_{\gamma^*}u\,\frac{\partial u}{\partial n}\,ds. \end{aligned}
\end{equation*}
\notag
$$
On the other hand3
$$
\begin{equation*}
I(u-u^*,B^*)=-\int_{\gamma^*}(u-u^*)\,\frac{\partial (u-u^*)}{\partial n}\,ds =\int_{\gamma^*}u^*\,\frac{\partial u}{\partial n}\,ds -\int_{ \gamma^*}u\,\frac{\partial u}{\partial n}\,ds.
\end{equation*}
\notag
$$
As a result, we obtain
$$
\begin{equation}
I(u^*,B^*)=I(u,B)-\int_{\gamma^*}u^*(z^*(s))\,\frac{\partial u}{\partial n^*}(z^*(s))\,ds^*.
\end{equation}
\tag{3.1}
$$
Our next aim is to replace the integral over $\gamma^*$ in (3.1) by an integral over $\gamma$. Let $\overline{n}^*$ denote the unit inward normal to $\gamma^*$ and the point $z^*(s)$ and $\overline{n}$ denote a similar normal to $\gamma$ at $z(s)$; let $\overline{l}^{\,*}$ be the unit tangent vector to $\gamma^*$ at $z^*(s)$ oriented in the positive direction of the boundary of $B^*$ and $\overline{l}$ be a similar tangent vector to $\gamma$ at $z(s)$. Let $\theta$ be the angle between $\overline{n}$ and $\overline{n}^*$. Then the direction cosines of $\overline{n}^*$ with respect to the vectors $\overline{l}$ and $\overline{n}$ have the form
$$
\begin{equation*}
\cos \alpha=\cos\biggl(\frac{\pi}{2}+\theta\biggr)=-\sin \theta\quad\text{and} \quad \cos \beta=\cos \theta.
\end{equation*}
\notag
$$
To find $\tan \theta$ we differentiate (1.1) with respect to the natural parameter $s$ on $\gamma$ and write
$$
\begin{equation*}
\frac{(z^*(s))'}{z'(s)}=1+i\varepsilon\biggl(\varphi'(s) +\varphi(s)\frac{z''(s)}{z'(s)}\biggr).
\end{equation*}
\notag
$$
Hence Therefore, uniformly in $s$ ($z(s)\in \gamma$).
Now we are ready to estimate the derivative $\partial u/\partial n^*$ uniformly in $s$:
$$
\begin{equation}
\begin{aligned} \, \notag \frac{\partial u}{\partial n^*}(z^*(s)) &=\frac{\partial u}{\partial l}(z^*(s))\cos\alpha+\frac{\partial u}{\partial n}(z^*(s))\cos\beta \\ \notag &=-\frac{\partial u}{\partial l}(z^*(s))\sin\theta+\frac{\partial u}{\partial n}(z^*(s))\cos\theta \\ &=-\frac{\partial u}{\partial l}(z^*(s))\varphi'(s)\varepsilon+\frac{\partial u}{\partial n}(z^*(s))+O(\varepsilon^2) \notag \\ &=-\frac{\partial u}{\partial l}(z(s))\varphi'(s)\varepsilon+\frac{\partial^2 u}{\partial n^2}(z(s))\varphi(s)\varepsilon+O(\varepsilon^2). \end{aligned}
\end{equation}
\tag{3.2}
$$
We show that uniformly in $s$. To do this it is sufficient to show that uniformly in $\overline{B}^*$. The function $v_{\varepsilon}(z)$ is continuous in $\overline{B}^*$, harmonic in $B^*$, vanishes on $\bigcup_{k=1}^n E_k$ and satisfies $\partial v_{\varepsilon}/\partial n=0$ on $(\partial B^*)\setminus(\bigcup_{k=1}^n E_k \cup \gamma^*)$. We let $z_{\varepsilon}$ denote a point in $\overline{B}^*$ at which $|v_{\varepsilon}|$ takes the maximum value in $\overline{B}^*$. Let $M_\varepsilon=|v_{\varepsilon}(z_\varepsilon)|$. By Hopf’s lemma this point lies on $\gamma^*$. Let $\zeta_{\varepsilon}$ be a point in the intersection $B\cap B^*$ such that for some $R>0$ the disc $|z-\zeta_{\varepsilon}|<R$ lies entirely in $B^*$ and ${|z_{\varepsilon}-\zeta_{\varepsilon}|=R}$. We can assume that $R$ is independent of the position of $z_{\varepsilon}$ on the curve $\gamma^*$. For definiteness let $v_{\varepsilon}(z_{\varepsilon})>0$. To the function $M_{\varepsilon}-v_{\varepsilon}$ and an arbitrary point $z'\in (z_{\varepsilon},\zeta_{\varepsilon})$ we apply Harnack’s inequality
$$
\begin{equation*}
M_{\varepsilon}-v_{\varepsilon}(\zeta_{\varepsilon})\leqslant (M_{\varepsilon}-v_{\varepsilon}(z'))\frac{|z_\varepsilon-\zeta_\varepsilon| +|z'-\zeta_\varepsilon|}{|z_\varepsilon-\zeta_\varepsilon|-|z'-\zeta_\varepsilon|}\leqslant 2R\frac{v_\varepsilon(z_\varepsilon)-v_\varepsilon(z')}{|z_\varepsilon-z'|}.
\end{equation*}
\notag
$$
Letting $z'$ tend to $z_\varepsilon$ we obtain
$$
\begin{equation}
|v_\varepsilon(\zeta_\varepsilon)|=v_\varepsilon(\zeta_\varepsilon)\geqslant M_\varepsilon-2R\biggl|\frac{\partial v_\varepsilon}{\partial n^*}(z_\varepsilon)\biggr|= M_\varepsilon-2R\biggl|\frac{\partial u}{\partial n^*}(z_\varepsilon)\biggr|=M_\varepsilon+O(\varepsilon).
\end{equation}
\tag{3.5}
$$
In a similar way we can prove the final estimate in (3.5) for $|v_\varepsilon(\zeta_\varepsilon)|=-v_\varepsilon(\zeta_\varepsilon)$.
Now consider the integral representation
$$
\begin{equation}
v_\varepsilon(\zeta_\varepsilon)=-\int_{\partial B^*} \biggl[\sigma(z,\zeta_\varepsilon)\,\frac{\partial v_\varepsilon}{\partial n}(z)-v_\varepsilon(z)\,\frac{\partial\sigma(z,\zeta_\varepsilon)}{\partial n}\biggr]\,|dz|.
\end{equation}
\tag{3.6}
$$
Here $\sigma(z,\zeta)$ is an arbitrary normalized fundamental solution of Laplace’s equation for the domain $B^*$ (see [15], pp. 255–257). For sufficiently small $\varepsilon>0$ we can take as $\sigma(z,\zeta)$ the function
$$
\begin{equation*}
\frac{1}{2\pi}g_B\biggl(z,\zeta,\bigcup_{k=1}^n E_k\biggr),
\end{equation*}
\notag
$$
which is $z$-continuous on $\overline{B}\setminus\{\zeta\}$, harmonic in $\overline{B}\setminus(\bigcup_{k=1}^n E_k\cup \{\zeta\})$, equal to zero on $\bigcup_{k=1}^n E_k$, has the zero normal derivative on $( \partial B)\setminus\bigcup_{k=1}^n E_k$ and has a singularity of the type of $-\frac{1}{2\pi}\log|z-\zeta|$ in a neighbourhood of $\zeta$ (see [4], § 2, [10], § 2.1, and also § 4 below). It follows from (3.6) and the preceding calculations that for points $\zeta_\varepsilon$ in a sufficiently small neighbourhood of an accumulation point of the set $\{\zeta_\varepsilon\}_{\varepsilon>0}$ we have
$$
\begin{equation*}
|v_\varepsilon(\zeta_\varepsilon)|\leqslant\int_{\gamma^*} \biggl[\,\biggl|\sigma(z,\zeta_\varepsilon)\,\frac{\partial u}{\partial n^*}(z)\biggr|+\biggl|v_\varepsilon(z)\,\frac{\partial\sigma(z,\zeta_\varepsilon)}{\partial n}\biggr|\,\biggr]\,|dz|\leqslant O(\varepsilon)+M_\varepsilon O(\varepsilon).
\end{equation*}
\notag
$$
Taking (3.5) into account we see that $M_\varepsilon=O(\varepsilon)$. Hence we have (3.4) and (3.3). Summing (3.1)–(3.3) and taking (1.1) into account we arrive at the variational formula
$$
\begin{equation*}
\operatorname{cap}C^*=\operatorname{cap}C+\varepsilon\int_{\gamma}u\biggl[\frac{\partial u} {\partial l}\varphi'(s)-\frac{\partial^2 u} {\partial n^2}\varphi(s)\biggr]\,ds+O(\varepsilon^2).
\end{equation*}
\notag
$$
We can deduce (1.3) from it by integrating by parts the component of the integrand containing $\varphi'(s)$ and by using the fact that $u$ is harmonic on $\gamma$.
Theorem 2 is proved. The above result is of particular interest in the case of multiply connected domains $B$ or condensers with many plates. On the other hand, if $B$ is simply connected (a quadrilateral), then the capacity of the condenser with two plates with levels of the potential on two opposite sides of $B$ equal to 0 and 1 is the reciprocal value to the capacity of the condenser in $B$ with plates on the other two sides and the same levels of the potential. Therefore, if the assumptions of Theorem 2 are fulfilled for the first condenser, then we can apply formula (1.2) to the second. § 4. Variations of quadratic formsIn what follows we use the notation from [10]. Let $B$ be a domain in the extended complex plane which is bounded by a finite number of analytic Jordan curves. Let $\Gamma$ be a nonempty closed subset of $\partial B$ formed by a finite number of nondegenerate Jordan arcs or closed curves, and let $\zeta$ be a finite point in $B$. We let $g_B(z,\zeta,\Gamma)$ denote the Robin function of $B$ with pole $\zeta$ (see [4]–[6]), that is, the continuous real function in $\overline{B}\setminus\{\zeta\}$ that is harmonic in $\overline{B}\setminus (\Gamma\cup\{\zeta\})$ and satisfies the following conditions:
$$
\begin{equation*}
\begin{gathered} \, g_B(z,\zeta,\Gamma)=0 \quad\text{for } z\in \Gamma, \\ \frac{\partial}{\partial n}g_B(z,\zeta,\Gamma)=0 \quad\text{for } z\in (\partial B)\setminus\Gamma, \end{gathered}
\end{equation*}
\notag
$$
and $g_B(z,\zeta,\Gamma)+\log|z-\zeta|$ is a bounded harmonic function in a neighbourhood of $\zeta$. For $\zeta=\infty$ the function $g_B(z,\infty,\Gamma)$ is defined in a similar way, except that now $g_B(z,\infty,\Gamma)-\log|z|$ must be harmonic in a neighbourhood of the point at infinity. Using conformal mappings we can extend the concept of Robin function to an arbitrary finitely connected domain without isolated boundary points in the complex plane. By the Robin radius of $B$ with respect to a point $\zeta$ and a set $\Gamma$ we mean the quantity
$$
\begin{equation*}
\begin{gathered} \, r(B,\Gamma,\zeta)=\exp\Bigl\{\lim_{z\to \zeta}[g_B(z,\zeta,\Gamma)+\log|z-\zeta|]\Bigr\}, \qquad \zeta \ne \infty, \\ r(B,\Gamma,\infty)=\exp\Bigl\{\lim_{z\to \infty}[g_B(z,\infty,\Gamma)-\log|z|]\Bigr\}. \end{gathered}
\end{equation*}
\notag
$$
In the case when $B$ is simply connected and $\Gamma$ is a boundary arc the quantity $r(B,\Gamma,\zeta)$ can be found in the literature under different names. For example, $1/r(B,\Gamma,\infty)$ is the same as the Robin capacity (for instance, see [6] and [9]). For $\Gamma=\partial B$ the Robin function is the Green’s function $g_B(z,\zeta)$ of the domain $B$ with pole $\zeta$, and the Robin radius is the inner radius4 $r(B,\zeta)$. It is natural to set Theorem 3. Let $B$ be a finitely connected domain in the plane $\overline{\mathbb{C}}$ which is bounded by Jordan curves, let $\{z_k\}_{k=1}^n$, $n\geqslant 1$, be a set of distinct points in $B$ and $\{\delta_k\}_{k=1}^n$ be a set of real numbers. Let $\Gamma$ be a nonempty closed subset of $\partial B$ which consists of a finite number of nondegenerate connected components, let $\gamma$ be a union of a finite number of closed analytic arcs or closed curves in $\mathbb{C}$ which lie on the boundary of $B$ and $\gamma^*$ be the union of curves obtained from $\gamma$ by a deformation of the form (1.1) such that the support of $\varphi(s)$ does not contain endpoints of $\gamma$; finally, let $B^*$ be the domain bounded by $((\partial B)\setminus\gamma)\cup \gamma^*$. If $\gamma\subset \Gamma$, then the following variational formula holds:
$$
\begin{equation}
\begin{aligned} \, \notag &\sum_{k=1}^n\sum_{l=1}^n\delta_k\delta_l g_{B^*}(z_k,z_l,(\Gamma\setminus \gamma)\cup\gamma^*) \\ &\qquad =\sum_{k=1}^n\sum_{l=1}^n\delta_k\delta_l g_B(z_k,z_l,\Gamma) -\frac{\varepsilon}{2\pi}\int_\gamma\biggl(\frac{\partial g}{\partial n}\biggr)^2\varphi(s)\,ds+O(\varepsilon^2). \end{aligned}
\end{equation}
\tag{4.1}
$$
For $\gamma\subset (\partial B)\setminus\Gamma$ the following formula holds:
$$
\begin{equation}
\begin{aligned} \, \notag &\sum_{k=1}^n\sum_{l=1}^n\delta_k\delta_l g_{B^*}(z_k,z_l,\Gamma) \\ &\qquad =\sum_{k=1}^n\sum_{l=1}^n\delta_k\delta_l g_B(z_k,z_l,\Gamma) +\frac{\varepsilon}{2\pi}\int_\gamma \biggl(\frac{\partial g}{\partial l}\biggr)^2\varphi(s)\,ds+O(\varepsilon^2). \end{aligned}
\end{equation}
\tag{4.2}
$$
Here $g(z)=\sum_{k=1}^n \delta_kg_B(z,z_k,\Gamma)$, $\partial/\partial n$ denotes differentiation along the normal to $\gamma$, and $\partial/\partial l$ denotes differentiation along the tangent to this curve. Proof. Using conformal mappings we can assume that the domain $B\subset \mathbb{C}$ is bounded by a finite number of analytic Jordan curves and the numbers $\delta_k$, $k=1,\dots,n$, are distinct from zero. For a sufficiently small $r>0$ we look at the function
$$
\begin{equation*}
\begin{aligned} \, &u(z) =-\frac{1}{\log r}\sum_{k=1}^n \delta_k g_B(z,z_k,\Gamma) \\ &\ -\frac{1}{(\log r)^2}\biggl\{\sum_{k=1}^n\delta_k[\log r(B,\Gamma,z_k)]g_B(z,z_k,\Gamma) +\sum_{k=1}^n\delta_k\sum_{{\substack{{l=1} \\ {l \neq k}}}}^n g_B(z_l,z_k,\Gamma)g_B(z,z_l,\Gamma)\biggr\} \end{aligned}
\end{equation*}
\notag
$$
and the sets
The function $u$ is continuous on $\overline{B}\setminus \bigcup_{k=1}^n\{z_k\}$, harmonic in $B\setminus \bigcup_{k=1}^n\{z_k\}$ and satisfies the following conditions: $\bullet$ $u=0$ on $\Gamma$; $\bullet$ $u=\delta_k$ on $\partial E(z_k,r)$, $k=1,\dots,n$; $\bullet$ $\partial u/\partial n=0$ on $(\partial B)\setminus \Gamma$ (see the proof of Theorem 2.2 in [10]). Thus, $u$ is the potential function of the condenser
$$
\begin{equation*}
C=(B,\{\Gamma,\partial E(z_1,r),\dots,\partial E(z_n,r)\},\{0,\delta_1,\dots,\delta_n\}).
\end{equation*}
\notag
$$
Assume that $\gamma\subset \Gamma$ and set
$$
\begin{equation*}
C^*=(B^*,\{(\Gamma\setminus\gamma)\cup \gamma^*,\partial E(z_1,r),\dots,\partial E(z_n,r)\},\{0,\delta_1,\dots,\delta_n\}).
\end{equation*}
\notag
$$
Repeating the proof of Theorem 1 word for word we verify that a formula of the form (1.2) must hold, so that
$$
\begin{equation}
\operatorname{cap}C^*=\operatorname{cap}C+\varepsilon\int_{\gamma}\biggl(\frac{\partial u}{\partial n}\biggr)^2\varphi(s)\,ds+O\biggl(\biggl(\frac{\varepsilon}{\log r}\biggr)^2\biggr)
\end{equation}
\tag{4.3}
$$
uniformly in $r$ and $\varepsilon$. By Theorem 2.2 in [10]
$$
\begin{equation*}
\begin{aligned} \, \operatorname{cap}C^* &=-\frac{2\pi}{\log r}\sum_{k=1}^n\delta_k^2 \\ &\qquad-\frac{2\pi}{(\log r)^2}\sum_{k=1}^n\sum_{l=1}^n\delta_k\delta_l g_{B^*}(z_k,z_l,(\Gamma\setminus \gamma)\cup\gamma^*)+o\biggl(\biggl(\frac{1}{\log r}\biggr)^2\biggr) \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{aligned} \, \operatorname{cap}C &=-\frac{2\pi}{\log r}\sum_{k=1}^n\delta_k^2-\frac{2\pi}{(\log r)^2}\sum_{k=1}^n\sum_{l=1}^n\delta_k\delta_l g_{B}(z_k,z_l,\Gamma)+o\biggl(\biggl(\frac{1}{\log r}\biggr)^2\biggr), \\ &\qquad\qquad\qquad\qquad\qquad r\to 0. \end{aligned}
\end{equation*}
\notag
$$
Substituting this into (4.3), after some simple manipulations we arrive at (4.1).
Now let $\gamma\subset (\partial B)\setminus\Gamma$. Then we look at the condenser
$$
\begin{equation*}
C^*=(B^*,\{\Gamma,\partial E(z_1,r),\dots,\partial E(z_n,r)\},\{0,\delta_1,\dots,\delta_n\}).
\end{equation*}
\notag
$$
Repeating the proof of Theorem 2 we arrive at an equality of the form (1.3) with $u$ as defined above and with the bound $O(({\varepsilon}/{\log r})^2)$ (in place of $O(\varepsilon^2)$) holding uniformly in $r$ and $\varepsilon$. We complete the proof by using Theorem 2.2 from [10], just as in the proof of formula (4.1).
Theorem 3 is proved. Various versions of inequality (4.1) for the simply connected domain $B$, $\Gamma=\partial B$ and $n=1$ are well known in the literature (for instance, see [2], § 63, [3], § 12, or [16], Lemmas 6 and 7). In terms of conformal mappings, (4.1) presents a variational formula for the modulus of the derivative at interior and boundary points of the domain of definition alike. For $\Gamma=\partial B$, (4.1) is a consequence of Hadamard’s variational formula (see [1], (A.3.3)). For $\Gamma\ne \partial B$, $n=1$ and $z_1=\infty$ formulae (4.1) and (4.2) describe variations of Robin capacity. For a simply connected domain $B$ this case was thoroughly examined by Nasyrov in terms of conformal mappings (see [9], Theorems 5 and 6). In addition, [9] contains applications of the results obtained there to estimates for variations of the aerodynamical lift of an airfoil. In the general case the quadratic forms in Theorem 3 were also considered in [4] and [12]. In [17] the reader can find general inequalities for such quadratic forms, which contain reduced moduli of strips and half-strips, with applications to problems of extremal partition and related problems in geometric function theory. AcknowledgementThe author is grateful to L. V. Kovalev for important information used in this work and to the referees for their useful comments. 
Citation in format AMSBIB
\Bibitem{Dub24} |
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