# Numerical solution of the heat conduction problem with memory111The
publication was financially supported by the research grant 20-01-00207 of
Russian Foundation for Basic Research

P.N. Vabishchevich22footnotemark: 233footnotemark: 3<EMAIL_ADDRESS>Nuclear Safety Institute, Russian Academy of Sciences, 52, B. Tulskaya, 115191
Moscow, Russia  North-Caucasus Center for Mathematical Research, North-
Caucasus Federal University, 1, Pushkin Street, 355017 Stavropol, Russia

###### Abstract

It is necessary to use more general models than the classical Fourier heat
conduction law to describe small-scale thermal conductivity processes. The
effects of heat flow memory and heat capacity memory (internal energy) in
solids are considered in first-order integrodifferential evolutionary
equations with difference-type kernels. The main difficulties in applying such
nonlocal in-time mathematical models are associated with the need to work with
a solution throughout the entire history of the process. The paper develops an
approach to transforming a nonlocal problem into a computationally simpler
local problem for a system of first-order evolution equations. Such a
transition is applicable for heat conduction problems with memory if the
relaxation functions of the heat flux and heat capacity are represented as a
sum of exponentials. The correctness of the auxiliary linear problem is
ensured by the obtained estimates of the stability of the solution concerning
the initial data and the right-hand side in the corresponding Hilbert spaces.
The study’s main result is to prove the unconditional stability of the
proposed two-level scheme with weights for the evolutionary system of
equations for modeling heat conduction in solid media with memory. In this
case, finding an approximate solution on a new level in time is not more
complicated than the classical heat equation. The numerical solution of a
model one-dimensional in space heat conduction problem with memory effects is
presented.

###### keywords:

media with memory , heat conduction equation , Volterra integrodifferential
equation , system of evolutionary equations , two-level schemes

###### MSC:

35K05 , 34K30 , 35R20 , 47G20 , 65J08 , 65M12

††journal: arXiv

##  1 Introduction

Classical thermal conduction is based on the Fourier heat conduction law,
where the heat flux is proportional to the temperature gradient. In this
approximation, the rate of heat propagation is infinite. When considering heat
transfer processes on small scales, we are forced to use more complex heat
transfer models [9, 17]. A classic example of non-Fourier heat conduction
equations is the hyperbolic heat conduction model when the effects of heat
flux relaxation are taken into account. The mathematical model becomes more
complicated but remains local: we have a second-order evolution equation for
temperature or a system of first-order equations for temperature and heat
flux. In the general case, we should focus on nonlocal heat transfer models,
where the thermal state depends on prehistory. Memory effects of heat flow and
heat capacity are accounted for using nonlocal evolution equations for
temperature [7, 13] with integral terms with difference-type kernels.

The problems of numerical solution of nonlocal in time IBVPs are well known.
We can use some or other quadrature formulas to approximate the integral time
terms and have suitable properties on stability and convergence of the
approximate solution to the exact one (see, for example, [4]). The main
problems are related to the computational cost of finding the solution on the
new time level. We have to operate with approximate solutions for all prior
time moments. Because of this, we are particularly interested in approaches
that allow us to go from a nonlocal model to a local model not much more
complex than the classical heat conduction equation.

We use in our work a well-known method of approximate solution of the Volterra
integral equations [12], which is associated with an approximation of the
kernel by a sum of exponents. The construction of such approximations is an
independent nonlinear approximation problem [1]. The justification of such a
transition for evolutionary first-order integrodifferential equations with
solution memory and solution derivative in time is given in our works [19,
18]. The present paper considers evolution equations for heat conduction
problems with heat flow memory and heat capacity memory.

The paper is organized as follows. In Section 2, we describe nonlocal heat
conduction problems with heat flow memory and heat capacity memory. In Section
3, we discuss the transition from a nonlocal problem with heat flow and energy
distribution functions as a sum of exponents to the Cauchy problem for a local
system of first-order evolution equations. Section 4 is devoted to the
construction and study of two-level operator-difference schemes. In Section 5,
we present examples of numerical solutions of one-dimensional spatial heat
conduction problems with memory. The final section is the conclusion.

##  2 Nonlocal heat conduction problems

We limit ourselves to considering linear models of heat conduction of a solid
isotropic body. In classical heat conduction theory, the heat flow
$\bm{q}(\bm{x},t)$ at a point $\bm{x}$ at some time $t$ is determined by the
temperature gradient $T(\bm{x},t)$ (Fourier’s law of heat conduction) when

$\bm{q}=-k\mathop{\rm grad}\nolimits T,$ (2.1)

where $k=\mathop{\rm const}\nolimits$ is the thermal conductivity. For a rigid
conductor the internal energy equation is

$\frac{\partial e}{\partial t}+\mathop{\rm div}\nolimits\bm{q}=r,$ (2.2)

where $r(\bm{x},t)$ is the distributed heat suction. The energy increments
$e(\bm{x},t)$ are due to changes in temperature, so that

$e=c\,T+\mathop{\rm const}\nolimits,$ (2.3)

where $c=\mathop{\rm const}\nolimits$ is the heat capacity. Substituting
(2.1), (2.3) into (2.2) gives the heat conduction equation

$c\frac{\partial T}{\partial t}-k\mathop{\rm div}\nolimits\mathop{\rm
grad}\nolimits T=r.$ (2.4)

Mathematical models of heat conduction with memory are proposed in [7, 13].
The prehistory of the process is accounted in [7] by specifying the heat flow
by the relation

$\bm{q}(\bm{x},t)=-\int_{0}^{t}\alpha(t-s)\mathop{\rm grad}\nolimits
T(\bm{x},s)ds,$ (2.5)

where $\alpha(t)$ is a positive, decreasing function relaxation of the heat
flux. In the special case $\alpha(t)=k\delta(t)$ ($\delta(t)$ is a
$\delta$-function), the Fourier law follows from (2.1), (2.5). The relation
(2.5) leads us instead of the equation (2.4) to the evolutionary
integrodifferential equation

$c\frac{\partial T}{\partial t}-\int_{0}^{t}\alpha(t-s)\mathop{\rm
div}\nolimits\mathop{\rm grad}\nolimits T(\bm{x},s)ds=r.$ (2.6)

Let’s assume that

$\alpha(t)=\frac{k}{\tau_{q}}\exp\Big{(}-\frac{t}{\tau_{q}}\Big{)},$ (2.7)

where $\tau_{q}$ is the relaxation time. In this case, from (2.5) we have the
Maxwell-Cattaneo law [3]

$\tau_{q}\frac{\partial\bm{q}}{\partial t}+\bm{q}=-k\mathop{\rm grad}\nolimits
T$ (2.8)

for the fluxes. This leads us to the hyperbolic heat conduction equation [2]

$c\tau_{q}\frac{\partial^{2}T}{\partial t^{2}}+c\frac{\partial T}{\partial
t}-k\mathop{\rm div}\nolimits\mathop{\rm grad}\nolimits
T=r+\tau_{q}\frac{\partial r}{\partial t}.$ (2.9)

The fundamental point is that instead of a nonlocal model (2.5) for the
exponential dependence of the relaxation function on time (2.7), we have a
local model (at each individual point $(\bm{x},t)$) for the dependence of heat
flux on temperature gradient as a differential relation (2.8). Instead of the
integrodifferential equation (2.6), a second-order evolution equation (2.9) is
used.

For the heat flux relaxation function we will use [13] the expression

$\alpha(t)=k\delta(t)+\tilde{\alpha}(t).$ (2.10)

Similar relations are used for internal energy relaxation when

$e(\bm{x},t)=\int_{0}^{t}\beta(t-s)T(\bm{x},s)ds+\mathop{\rm const}\nolimits,$
(2.11) $\beta(t)=c\delta(t)+\tilde{\beta}(t).$ (2.12)

From (2.2), (2.5), (2.10)–(2.12), we obtain the heat conduction equation with
memory

$\begin{split}c\frac{\partial T}{\partial t}&+\frac{\partial}{\partial
t}\int_{0}^{t}\tilde{\beta}(t-s)T(\bm{x},s)ds\\\ &-k\mathop{\rm
div}\nolimits\mathop{\rm grad}\nolimits
T-\int_{0}^{t}\tilde{\alpha}(t-s)\mathop{\rm div}\nolimits\mathop{\rm
grad}\nolimits T(\bm{x},s)ds=r.\end{split}$ (2.13)

We consider a numerical solution of the boundary value problem for the
equation (2.13). For simplicity, we restrict ourselves to the case of a
bounded domain $\Omega$ with a boundary $\partial\Omega$ on which homogeneous
Dirichlet boundary conditions are given:

$T(\bm{x},t)=0,\quad\bm{x}\in\partial\Omega.$ (2.14)

The initial thermal state is defined according to

$T(\bm{x},0)=T_{0}(\bm{x}),\quad\bm{x}\in\partial\Omega.$ (2.15)

Single-valued solvability of boundary value problems of type (2.13)–(2.15) in
the corresponding functional classes, the qualitative properties of the
solutions are discussed, for example, in [5, 6].

The object of our consideration is computational algorithms for approximate
solutions of heat conduction problems with allowance for memory effects. The
main problems are generated by the nonlocality of the equation (2.13), the
fact that the solution depends on the complete prehistory of the process. When
finding the solution at a new level in time, we have to consider the solution
at previous levels in time, which, in particular, is associated with high
computational and memory costs. Distinct possibilities of a principal
reduction of computational work are seen on the example of the transition to
the local equation (2.9) due to transition from the integrodifferential
relation (2.5) to the differential relation (2.8) when the heat flow
distribution function is given as (2.7). We can expect a similar
simplification of the problem when the heat flux and energy distribution
functions are represented as

$\tilde{\alpha}(t)=\sum_{i=1}^{m}\alpha_{i}\exp(-\mu_{i}t),\quad\tilde{\beta}(t)=\sum_{j=1}^{l}\beta_{j}\exp(-\nu_{j}t).$
(2.16)

For general dependencies $\tilde{\alpha}(t),\tilde{\beta}(t)$ this
representation corresponds to using nonlinear approximations of functions by
the sum of exponents [1].

##  3 Local system of equations

After a finite-element or finite-volume approximation over the space, we
consider the discrete problem in the corresponding finite-dimensional Hilbert
space $H$ [10, 14]. For the scalar product and norm we use the notation
$(u,v)$ and $\|u\|=(u,u)^{1/2}$ for $u,v\in H$ respectively. A positively
determined selfadjoint operator $D$ is associated with the Hilbert space
$H_{D}$, in which $(u,v)_{D}=(Du,v)$, $\|u\|_{D}=(Du,u)^{1/2}$.

The dimensionless problem (2.13)–(2.15) leads us to the equation

$\frac{du}{dt}+\frac{d}{dt}\int_{0}^{t}\tilde{\beta}(t-s)u(s)ds+Au+\int_{0}^{t}\tilde{\alpha}(t-s)Au(s)ds=f(t),\quad
t>0,$ (3.1)

for $u(\cdot,t)=u(t)$. The operator $A$ is associated with an approximation of
the Laplace operator for functions satisfying boundary conditions (2.14). It
is natural to assume that in $H$ it is constant (independent of $t$) and
$A=A^{*}>0$. We consider the Cauchy problem for the Volterra evolutionary
equation (3.1) when we use the initial condition

$u(0)=u_{0}.$ (3.2)

The transition from a nonlocal to a local problem when approximating the
difference-type kernel by a sum of exponents is provided by introducing new
desired variables. For the problem (2.16), (3.1), (3.2), we put

$v_{i}(t)=\int_{0}^{t}\exp(-\mu_{i}(t-s))u(s)ds,\quad i=1,2,\ldots,m,$
$w_{j}(t)=\int_{0}^{t}\exp(-\nu_{j}(t-s))u(s)ds,\quad j=1,2,\ldots,l.$

For these auxiliary functions, we have the local relations

$\frac{dv_{i}}{dt}+\mu_{i}v_{i}-u(t)=0,\quad i=1,2,\ldots,m,$ (3.3)
$\frac{dw_{j}}{dt}+\nu_{j}w_{j}-u(t)=0,\quad j=1,2,\ldots,l.$ (3.4)

Equation (3.1) is written as

$\frac{du}{dt}+\sum_{j=1}^{l}\beta_{j}\frac{dw_{j}}{dt}+Au+\sum_{i=1}^{m}\alpha_{i}Av_{i}=f(t).$

Given (3.4), we have

$\frac{du}{dt}+\sum_{j=1}^{l}\beta_{j}u+Au-\sum_{j=1}^{l}\beta_{j}\nu_{j}w_{j}+\sum_{i=1}^{m}\alpha_{i}Av_{i}=f(t).$
(3.5)

The initial condition (3.2) for $u(t)$ is supplemented by initial conditions
for auxiliary functions:

$v_{i}(0)=0,\quad i=1,2,\ldots,m,\quad w_{j}(0)=0,\quad j=1,2,\ldots,l.$ (3.6)

Thus from the nonlocal problem (2.16), (3.1), (3.2) for $u(t)$, we arrive at
the local Cauchy problem for the system of equations (3.2)–(3.6).

The main result concerns the stability for the initial data and the right-hand
side for the system of equations (3.3)–(3.5). The correctness of this linear
problem is established under the assumption that the coefficients in the
representations (2.16) are positive:

$\alpha_{i}>0,\quad\mu_{i}>0,\quad
i=1,2,\ldots,m,\quad\beta_{j}>0,\quad\nu_{j}>0,\quad j=1,2,\ldots,l.$ (3.7)

######  Theorem 1.

If (3.7) is satisfied, for the solution of the Cauchy problem (3.2)–(3.6),
there is an a priori estimate

$\|y(t)\|_{*}\leq\|u_{0}\|+\int_{0}^{t}\|f(s\|ds,$ (3.8)

where

$\|y(t)\|_{*}^{2}=\|u(t)\|^{2}+\sum_{i=1}^{m}\alpha_{i}\|v_{i}(t)\|_{A}^{2}+\sum_{j=1}^{l}\beta_{j}\nu_{j}\|w_{j}(t)\|^{2}.$

######  Proof 1.

Let us multiply equation (3.5) scalarly in $H$ by $u(t)$, separate equations
(3.3) by $\alpha_{i}Av_{i}(t)$, and equations (3.4) by
$\beta_{j}\nu_{j}w_{j}(t)$. The summation of these equations gives

$\begin{split}\frac{1}{2}&\frac{d}{dt}\Big{(}\|u(t)\|^{2}+\sum_{i=1}^{m}\alpha_{i}\|v_{i}(t)\|_{A}^{2}+\sum_{j=1}^{l}\beta_{j}\nu_{j}\|w_{j}|^{2}\Big{)}+\|u(t)\|_{A}^{2}\\\
&+\sum_{i=1}^{m}\alpha_{i}\mu_{i}\||v_{i}(t)\|_{A}^{2}+\sum_{j=1}^{l}\beta_{j}\|u(t)-\nu_{j}w_{j}\|^{2}=(f,u).\end{split}$
(3.9)

Given

$(f,u)\leq\|f(t)\|\|u\\\ |\leq\|f(t)\|\||y(t)\|_{*},$

from (3.9), we have the desired estimate (3.8).

From (3.8), a simpler estimate follows:

$\|u(t)\|\leq\|u_{0}\|+\int_{0}^{t}\|f(s\|ds.$

This a priori estimate is standard for the problem without memory effects
($\tilde{\alpha}(t)=\tilde{\beta}(t)\equiv 0$ in equation (3.1)).

##  4 Two-level difference scheme

When approximating first-order evolution equations, it is natural to focus on
standard two-level schemes with weights [11, 15]. Without limiting generality,
we will assume that the time grid is uniform, i.e. $t^{n}=n\tau,\
n=0,1,\ldots,$ where $\tau>0$ is time step. We denote the approximate solution
at time $t^{n}$ by $u^{n},\ \ n=0,1,\ldots$ and let

$t^{n+\sigma}=\sigma t^{n+1}+(1-\sigma)t^{n},\quad u^{n+\sigma}=\sigma
u^{n+1}+(1-\sigma)u^{n},$

where $\sigma=\mathop{\rm const}\nolimits,\ \sigma\in[0,1]$ is the weight
parameter.

For the approximate solution of the problem (3.2)–(3.6), we will use a two-
level scheme

$\frac{u^{n+1}-u^{n}}{\tau}+\sum_{j=1}^{l}\beta_{j}u^{n+\sigma}+Au^{n+\sigma}-\sum_{j=1}^{l}\beta_{j}\nu_{j}w_{j}^{n+\sigma}+\sum_{i=1}^{m}\alpha_{i}Av_{i}^{n+\sigma}=\varphi^{n+\sigma},$
(4.1)
$\frac{v_{i}^{n+1}-v_{i}^{n}}{\tau}+\mu_{i}v_{i}^{n+\sigma}-u^{n+\sigma}=0,\quad
i=1,2,\ldots,m,$ (4.2)
$\frac{w_{j}^{n+1}-w_{j}^{n}}{\tau}+\nu_{j}w_{j}^{n+\sigma}-u^{n+\sigma}=0,\quad
j=1,2,\ldots,l,\quad n=0,1,\ldots,$ (4.3) $u^{0}=u_{0},\quad v_{i}^{0}=0,\quad
i=1,2,\ldots,m,\quad w_{j}^{0}=0,\quad j=1,2,\ldots,l,$ (4.4)

where, for example, $\varphi^{n+\sigma}=f(t^{n+\sigma})$. The stability of
this scheme is established under standard constraints on the weight of
$\sigma$.

######  Theorem 2.

If (3.7) and $\sigma\geq 0.5$ are satisfied, the scheme (4.1)–(4.4) is
unconditionally stable and for the approximate solution the estimate

$\|y^{n+1}\|_{*}\leq\|u_{0}\|+2\sigma\sum_{k=1}^{n}\tau\|\varphi^{k+\sigma}\|,\quad
n=0,1,\ldots,$ (4.5)

where

$\|y^{n}\|_{*}^{2}=\|u^{n}\|^{2}+\sum_{i=1}^{m}\alpha_{i}\|v_{i}^{n}\|_{A}^{2}+\sum_{j=1}^{l}\beta_{j}\nu_{j}\|w_{j}^{n}\|^{2},$

is true.

######  Proof 2.

If we multiply equation (4.1) by $\tau u^{n+\sigma}$, equation (4.2) by
$\tau\alpha_{i}Av_{i}^{n+\sigma}$, and equation (4.3) by
$\tau\beta_{j}\nu_{j}w_{j}^{n+\sigma}$, we get the inequality

$\begin{split}((u^{n+1}&-u^{n}),u^{n+\sigma})+\sum_{i=1}^{m}\alpha_{i}(A(v_{i}^{n+1}-v_{i}^{n}),v_{i}^{n+\sigma})\\\
&+\sum_{j=1}^{l}\beta_{j}\nu_{j}((w_{j}^{n+1}-w_{j}^{n}),w_{j}^{n+\sigma})\leq\tau(\varphi^{n+\sigma},u^{n+\sigma})\end{split}$
(4.6)

Given that

$2u^{n+\sigma}=(2\sigma-1)(u^{n+1}-u^{n})+(u^{n+1}+u^{n}),$

at $\sigma\geq 0.5$, we have

$((u^{n+1}-u^{n}),u^{n+\sigma})\geq\frac{1}{2}(\|u^{n+1}\|^{2}-\|u^{n}\|^{2}).$

From the inequality (4.6), it follows

$\|y^{n+1}\|_{*}^{2}-\|y^{n}\|_{*}^{2}\leq
2\tau\||\varphi^{n+\sigma}\|\|u^{n+\sigma}\|.$ (4.7)

Under our constraints on $\sigma$, we have

$\|u^{n+\sigma}\|\leq\sigma(\|u^{n+1}\|+\|u^{n}\|)\leq\sigma(\|y^{n+1}\|_{*}+\|y^{n}\|_{*}).$

Thus from the inequality (4.7), we obtain the estimate

$\|y^{n+1}\|_{*}\leq\|y^{n}\|_{*}+2\sigma\tau\|\varphi^{n+\sigma}\|.$

From this follows the inequality being proved (4.5).

The computational realization of the scheme with weights (4.1)–(4.4) can be
organized as follows. From equations (4.2), (4.3), we have

$\begin{split}v_{i}^{n+1}&=\frac{\sigma\tau}{1+\sigma\mu_{i}\tau}u^{n+1}+\eta_{i}^{n},\\\
\eta_{i}^{n}&=\frac{1}{1+\sigma\mu_{i}\tau}\Big{(}(1-\sigma)\tau
u^{n}+\big{(}1-(1-\sigma)\mu_{i}\tau\big{)}v_{i}^{n}\Big{)},\quad
i=1,2,\ldots,m,\end{split}$ (4.8)
$\begin{split}w_{j}^{n+1}&=\frac{\sigma\tau}{1+\sigma\nu_{j}\tau}u^{n+1}+\theta_{j}^{n},\\\
\theta_{j}^{n}&=\frac{1}{1+\sigma\nu_{j}\tau}\Big{(}(1-\sigma)\tau
u^{n}+\big{(}1-(1-\sigma)\nu_{j}\tau\big{)}w_{j}^{n}\Big{)},\quad
j=1,2,\ldots,l.\end{split}$ (4.9)

Substituting expressions (4.8) and (4.9) into equation (4.1) gives

$(b_{I}I+\sigma\tau b_{A}A)u^{n+1}=\chi^{n},$ (4.10)

where $I$ is a unit operator, and

$b_{I}=1+\sigma\tau\sum_{j=1}^{l}\frac{\beta_{j}}{1+\sigma\nu_{j}\tau},\quad
b_{A}=1+\sigma\tau\sum_{i=1}^{m}\frac{\alpha_{i}}{1+\sigma\mu_{i}\tau}.$

For the right-hand side of equation (4.10), we have

$\begin{split}\chi^{n}&=\Big{(}1-(1-\sigma)\tau\big{(}\sum_{j=1}^{l}\beta_{j}+A\big{)}\Big{)}u^{n}+\tau\varphi^{n+\sigma}\\\
&+\tau\sum_{j=1}^{l}\beta_{j}\nu_{j}\big{(}\sigma\theta_{j}^{n}+(1-\sigma)w_{j}^{n}\big{)}-\tau\sum_{i=1}^{m}\alpha_{i}A\big{(}\sigma\eta_{i}^{n}+(1-\sigma)v_{i}^{n}\big{)}.\end{split}$

The operator $B=B^{*}>0$, where $B=b_{I}I+\sigma\tau b_{A}A$, and so equation
(4.10) is uniquely solvable for $u^{n+1}$. After finding $u^{n+1}$, the
auxiliary quantities $v_{i},\ i=1,2,\ldots,m,$ $w_{j},\ j=1,2,\ldots,l,$ are
calculated according to (4.8), (4.9). The increase in the computational
complexity of the numerical solution of problems with memory is not
fundamental in comparison to the problem without memory effects.

##  5 Numerical examples

In the above calculations, we have limited ourselves to one-dimensional
problems in space. Considering the current state of numerical solution of
boundary value problems for elliptic equations, the transition to more complex
multidimensional problems of heat conduction with memory is not of fundamental
nature. The methodological issues of estimating the rate of convergence of the
approximate solution to the exact solution of the system of evolution
equations are not discussed. We present numerical data on sufficiently
detailed computational grids in time and space when we can neglect
computational errors.

To illustrate the peculiarities of problems with memory effects, we consider
the equation (3.1) in which

$\tilde{\alpha}(t)=\alpha\exp(-\mu t),\quad\tilde{\beta}(t)=\beta\exp(-\nu
t).$

i.e., in (2.6) $m=l=1$. The system of equations (3.3)–(3.5) takes the form

$\frac{dv}{dt}+\mu v-u(t)=0,$ (5.1) $\frac{dw}{dt}+\nu w-u(t)=0,$ (5.2)
$\frac{du}{dt}+\beta u+Au-\beta\nu w+\alpha Av=f(t).$ (5.3)

For (5.1)–(5.3), a Cauchy problem is posed when

$u(0)=u_{0},\quad v(0)=0,\quad w(0)=0.$ (5.4)

It makes sense to distinguish the following separate classes of heat
conduction problems, for which we will give the corresponding equation for
temperature.

1\. Classical thermal conductivity.

The standard model of thermal conductivity without regard to memory effects
corresponds to the initial condition (3.2) and $\alpha=\beta=0$:

$\frac{du}{dt}+Au=f(t).$

2\. Heat flux memory

In this case $\alpha\neq 0,\ \ \beta=0$. From (5.1)–(5.3) we get

$\frac{d^{2}u}{dt^{2}}+(\mu+A)\frac{du}{dt}+(\mu+\alpha)Au=\frac{df}{dt}+\mu
f(t).$

In this telegraph equation, the term at the first derivative of the solution
in time describes the damping effects and the term with the solution — the
oscillatory effects. The initial conditions follow from (5.4) and equation
(5.3) at $t=0$:

$u(0)=u_{0},\quad\frac{du}{dt}=f(0)-Au_{0}.$

3\. Heat capacity memory

The heat capacity memory effects are accounted for by choosing $\alpha=0,\
\beta\neq 0$. From (5.1)–(5.3), we get

$\frac{d^{2}u}{dt^{2}}+(\beta+\nu+A)\frac{du}{dt}+\nu Au=\frac{df}{dt}+\nu
f(t).$

Again we have a telegraph equation for temperature, for which the initial
conditions are given as

$u(0)=u_{0},\quad\frac{du}{dt}=u_{1},\quad u_{1}=f(0)-(\beta+A)u_{0}.$

4\. General case

With $\alpha\neq 0,\ \ \beta\neq 0$ we have a third-order evolutionary
equation:

$\begin{split}\frac{d^{3}u}{dt^{3}}&+(\mu+\nu+\beta+A)\frac{d^{2}u}{dt^{2}}\\\
&+\big{(}(\mu+\nu)(\beta+A)+\nu(\mu-\beta)+\alpha
A\big{)}\frac{du}{dt}+\nu(\mu+\alpha)Au\\\
&=\frac{d^{2}f}{dt^{2}}+(\mu+\nu)\frac{df}{dt}+\mu\nu f(t).\end{split}$

The initial conditions (5.4) on the solutions of the system of equations
(5.1)–(5.3) are

$u(0)=u_{0},\quad\frac{du}{dt}=u_{1},\quad\frac{d^{2}u}{dt^{2}}=\frac{df}{dt}(0)-(\beta+A)u_{1}+(\beta\nu-\alpha
A)u_{0}.$

The system of equations (5.1)–(5.3) can be seen as one possible transition
from a third-order evolutionary equation to a first-order system of equations.

Numerical illustrations of the features of local generalized thermal
conductivity models are given, usually (see, for example, [16, 8, 20]) tracing
the dynamics of the initial temperature profile. We consider a similar test
problem whose numerical solution is carried out with a difference space
approximation. Let us assume that in dimensionless variables $x\in[0,1]$ and a
uniform grid with step $h$ is introduced. Let us denote by $\omega$ the set of
internal nodes of the grid in space. Given (2.14), for $y(x)=0,\
x\notin\omega$, we define the difference operator $A$ by the relation

$Ay=-\frac{1}{h^{2}}\big{(}y(x+h)-2y(x)+y(x-h)\big{)},\quad x\in\omega.$

The initial condition (3.2) is taken as

$u_{0}(x)=\left\\{\begin{array}[]{rr}x,&0<x\leq 0.5,\\\ \ 0,&0.5<x<1,\\\
\end{array}\right.\quad x\in\omega.$

The calculations were performed on a grid over a space with $h=2\cdot
10^{-3}$, with $0<t\leq T,\ T=0.1$, and $\tau=5\cdot 10^{-5}$. A purely
implicit scheme was used ($\sigma=1$ in the scheme (4.1)–(4.4)).

The dynamics of the process when using model 1 is presented in Fig. 1, and for
model 4 — in Fig. 2. The effects of heat flux memory and heat capacity lead to
a significant rearrangement of the temperature profile. The influence of
individual parameters of the mathematical model is traced to models 2 and 3.

Figure 1: Solution for individual levels in time: model 1 ($\alpha=\beta=0$).
Figure 2: Solution for individual levels in time: model 4 ($\alpha=\beta=5$,
$\mu=\nu=1$).

The results of heat flow memory (model 2) are presented in Figs. 3, 4. The
influence of the parameter $\alpha$ is most pronounced. At large $\alpha$
strong oscillatory effects are observed. In particular, at time $t=0.1$ at
$\alpha=10$, we observe negative values of the solution $u(x,t)$.

Figure 3: Effect of the parameter $\mu$: solution at $t=0.1$, model 2
($\beta=0$) at $\alpha=5$. Figure 4: Effect of the parameter $\alpha$:
solution at $t=0.1$, model 2 ($\beta=0$) at $\mu=1$.

Similar calculated data for heat capacity memory (model 3) are presented in
Figs. 5, 6. In this model, the effects of dissipation are most pronounced.

Figure 5: Effect of parameter $\nu$: solution at $t=0.1$, model 3 ($\alpha=0$)
at $\beta=5$. Figure 6: Effect of parameter $\beta$: solution at $t=0.1$,
model 3 ($\alpha=0$) at $\nu=1$.

##  6 Conclusions

  1. 1.

A boundary value problem for the first order integrodifferential evolution
equation with difference-type kernels, which describes heat conduction
processes in solids in linear approximation with allowance for memory effects,
is formulated. Processes with a memory of heat flow and memory of heat
capacity (internal energy) are modeled.

  2. 2.

The nonlocal time problem is reduced to a system of local evolution equations
of the first order under the condition that the heat flow and heat capacity
relaxation functions are represented as a sum of exponents. We obtained an
estimate of the stability of the solution of the Cauchy problem for the system
of equations concerning the initial data and the right-hand side in the
corresponding Hilbert spaces.

  3. 3.

A unconditionally stable scheme with weights for an evolutionary system of
equations modeling thermal conductivity in solid media with memory has been
proposed and investigated. The transition to a new level in time is not more
complicated than the computational implementation of standard schemes with
weights for the classical heat conduction equation.

  4. 4.

Numerical results of using a purely implicit scheme in approximate solutions
of heat conduction problems with allowance for memory effects are presented.
The evolution of the initial state at different values of model parameters is
considered.

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# A note on the squeezing function

Alexander Yu. Solynin  Department of Mathematics and Statistics, Texas Tech
University, Box 41042, Lubbock, Texas 79409<EMAIL_ADDRESS>
###### Abstract.

The squeezing problem on $\mathbb{C}$ can be stated as follows. Suppose that
$\Omega$ is a multiply connected domain in the unit disk $\mathbb{D}$
containing the origin $z=0$. How far can the boundary of $\Omega$ be pushed
from the origin by an injective holomorphic function $f:\Omega\to\mathbb{D}$
keeping the origin fixed?

In this note, we discuss recent results on this problem obtained by Ng, Tang
and Tsai (Math. Anal. 2020) and by Gumenyuk and Roth (arXiv:2011.13734, 2020)
and also prove few new results using a method suggested in one of our previous
papers (Zapiski Nauchn. Sem. POMI 1993).

###### Key words and phrases:

Squeezing function, circularly slit disk, doubly connected domain, Jenkins’s
module problem

###### 2010 Mathematics Subject Classification:

Primary 30C75; Secondary 30C35

The squeezing problem. The squeezing function $S_{\Omega}(z)$ of a planar
domain $\Omega$ is defined as follows. Suppose that $\Omega\subset\mathbb{C}$
is such that there is an injective holomorphic function $f(z)$ from $\Omega$
to the unit disk $\mathbb{D}=\\{z:\,|z|<1\\}$. Let $\mathcal{U}(\Omega)$ be
the class of all such functions. Then $S_{\Omega}:\Omega\to\mathbb{R}$ is
defined by

$S_{\Omega}(z)=\sup\\{{\mbox{dist}}(0,\partial
f(\Omega)):\,f\in\mathcal{U}(\Omega),\,f(z)=0\\}.$ (1)

According to [8], the squeezing function was first introduced in 2012 by Dong,
Guan and Zhang [8] but the concept itself goes back to the work of Liu, Sun
and Yau [7]. These authors defined $S_{\Omega}$ and used it in a more general
setting, namely, for classes of injective holomorphic mappings defined on
domains $\Omega$ in $\mathbb{C}^{n}$, $n\geq 1$.

The _squeezing problem_ , i.e. the problem to find or characterize
$S_{\Omega}(z)$, is a difficult task, even in complex dimension $1$. Any
function $f\in\mathcal{U}(\Omega)$ such that $f(z)=0$ and
$S_{\Omega}(z)={\mbox{dist}}(0,\partial f(\Omega))$ will be called extremal
for the squeezing problem (for the point $z\in\Omega$). In this case, the
image $f(\Omega)$ will be called an extremal domain.

For doubly connected domains $\Omega\subset\mathbb{C}$, the squeezing function
was identified by Ng, Tang and Tsai in 2020 [8]. To state the main result of
[8], we first introduce necessary notations. As well known, see [4, Theorem
5.4], every domain $\Omega\subset\mathbb{C}$ with a finite number of boundary
continua $\gamma_{1},\ldots,\gamma_{n}$, $n\geq 2$, can be mapped by
$w=\varphi_{a,k}(z)$, $a\in\Omega$, $1\leq k\leq n$, conformally on
$\mathbb{D}$ slit along arcs $C_{j,k}=C_{j,k}(a)$, $j\not=k$, on the circles
of radii $0<r_{j,k}=r_{j,k}(a)<1$ centered at $0$ so that
$\varphi_{a,k}(a)=0$, a non-degenerate boundary continuum $\gamma_{k}$
corresponds to the unit circle $\mathbb{T}=\partial\mathbb{D}$ and, for
$j\not=k$, $\gamma_{j}$ corresponds to $C_{j,k}$. Under the additional
normalization $\varphi^{\prime}_{a,k}(a)>0$, that we assume in what follows,
the mapping function $\varphi_{a,k}$ is uniquely determined. If $\gamma_{k}$
is a singleton, we put $r_{j,k}(a)=0$ for all $a\in\Omega$ and all $j\not=k$.
Without the uniqueness statement, the mapping property, discussed above,
remains true for domains of any connectivity.

######  Theorem 1 ([8],[3]).

Let $\Omega\subset\mathbb{C}$ be a doubly connected domain with boundary
components $\gamma_{1}$ and $\gamma_{2}$, at least one of which is non-
degenerate. Then

$S_{\Omega}(z)=\max\\{r_{1,2}(z),r_{2,1}(z)\\}.$ (2)

If $r_{1,2}(z)<r_{2,1}(z)$, then $\varphi_{z,1}$ is the unique (up to rotation
about the origin) extremal function for the squeezing problem; if
$r_{1,2}(z)>r_{2,1}(z)$, then $\varphi_{z,2}$ is the unique extremal function;
and if $r_{1,2}(z)=r_{2,1}(z)$, then both $\varphi_{z,1}$ and $\varphi_{z,2}$
are extremal.

This theorem, except for the uniqueness part, was proved in [8] using the
Loewner differential equation and tricky calculations with special functions.
A simpler proof based on the potential theory, which also includes the proof
of the uniqueness statement, was presented in [3].

In this note, we first show that Theorem 1 is immediate from Theorem 1.2 in
our 1993 paper [10]. Then we discuss how the method used in [10] can be
applied to study the squeezing problem in a more general setting.

Let $0<r_{1}\leq r_{2}<1$ and let $E$ be a compact subset in the closure
$\bar{A}(r_{1},r_{2})$ of the annulus
$A(r_{1},r_{2})=\\{z:\,r_{1}<|z|<r_{2}\\}$. Let
$\mathcal{D}(E)=\\{(D_{1},D_{2})\\}$ be the set of pairs $(D_{1},D_{2})$ of
non-overlapping domains in $\mathbb{D}\setminus E$, where $D_{1}$ is a simply
connected domain containing the origin and $D_{2}$ is a doubly connected
domain separating $E$ from $\mathbb{T}$. In what follows, $m(D,z_{0})$ stands
for the reduced module of a simply connected domain $D$ with respect to the
point $z_{0}\in D$ and $m(D)$ stands for the module of a doubly connected
domain $D$. For the definitions and properties of these moduli, we refer to
Jenkins’s monograph [4] as the primary source and also to [6], [2], and [11] .
Figure 1 illustrates our notations introduced above.

Consider the following _module problem_.

Problem M. _Given nonnegative numbers $\alpha_{1}$ and $\alpha_{2}$, at least
one of which is positive, identify all pairs
$(D_{1}^{*},D_{2}^{*})\in\mathcal{D}(E)$, which maximize the weighted sum of
moduli_

$\alpha_{1}^{2}m(D_{1},0)+\alpha_{2}^{2}m(D_{2})$ (3)

_over the set $\mathcal{D}(E)$._

######  Theorem 2 ([10, Theorem 1.2]).

(1) There is a unique pair $(D_{1}^{*},D_{2}^{*})\in\mathcal{D}(E)$ maximizing
the sum (3) over the class $\mathcal{D}(E)$. The domains $D_{1}^{*}$ and
$D_{2}^{*}$ are, respectively, a circle domain and a ring domain of a
quadratic differential $Q(z)\,dz^{2}$ defined on $\mathbb{D}\setminus E$,
which is positive on $\mathbb{T}$ and has a second order pole with circular
local structure of trajectories at $z=0$.

(2) Let $L(E)$ denote the _free boundary_ of the module problem; i.e.
$L(E)=(\partial D_{1}^{*}\cup\partial D_{2}^{*})\cap(\mathbb{D}\setminus E)$.
Then $L(E)$ consists of arcs of critical trajectories of $Q(z)\,dz^{2}$ and
their endpoints in $\mathbb{D}\setminus E$.

(3) The following holds: (a) if $\alpha_{1}\leq\alpha_{2}$, then
$L(E)\subset\bar{\mathbb{D}}_{r_{2}}$, where $\mathbb{D}_{r}=\\{z:\,|z|<r\\}$;
(b) if $\alpha_{1}\geq\alpha_{2}$, then $L(E)\subset\bar{A}(r_{1},1)$; (c) if
$\alpha_{1}=\alpha_{2}$, then $L(E)\subset\bar{A}(r_{1},r_{2})$.

We note that Problem M is a particular case of Jenkins’s problem on extremal
partitioning of Riemann surfaces, see [5], [6], [11]. Therefore, parts (1) and
(2) of this theorem follow from Jenkins’s theorem on extremal partitioning;
see [5, Theorem 1]. Part (3), which is essential for this paper, was proved in
[10]. For convenience of the readers and because this result seems useful in
the study of the squeezing function (1), a version of this proof will be
presented at the end of this note.

Figure 1. Extremal partitioning into two domains.

_Proof of Theorem 1._ Using an auxiliary conformal mapping, we may assume
without loss of generality that $\Omega$ is the unit disk $\mathbb{D}$ slit
along a proper arc $C_{1,2}$ of the circle $\\{z:\,|z|=r_{1,2}\\}$, that
$z=0$, and that $r_{1,2}\geq r_{2,1}$. Consider Problem M with
$\alpha_{1}=\alpha_{2}=1$ for the set $E=C_{1,2}$. We call it Problem $P_{1}$.
It follows from Grötzsch’s lemma [4, Theorem 2.6] (or from Theorem 2(3) above)
that $(\mathbb{D}_{r_{1,2}},A(r_{1,2},1))$ is the extremal pair of domains for
Problem $P_{1}$ and the complementary arc
$C^{\prime}_{1,2}=\\{z:\,|z|=r_{1,2}\\}\setminus C_{1,2}$ is the free boundary
of Problem $P_{1}$.

Now, suppose, by contradiction, that $S_{\Omega}(0)>r_{1,2}$. The latter
implies that there is a function $f\in\mathcal{U}(\mathbb{D}\setminus
C_{1,2})$ such that $f(0)=0$ and ${\mbox{dist}}(0,f(C_{1,2}))>r_{1,2}$. Let
$\rho_{1}=\inf\\{|z|:\,z\in f(C_{1,2})\\}$, $\rho_{2}=\sup\\{|z|:\,z\in
f(C_{1,2})\\}$. Consider Problem M, again with $\alpha_{1}=\alpha_{2}=1$, for
the set $f(C_{1,2})\subset\bar{A}(\rho_{1},\rho_{2})\\}$. We call it Problem
$P_{2}$. Since Jenkins’s problem on extremal partitioning is conformally
invariant (in the sense that conformal mappings preserve extremal
configurations), it follows that the pair
$(f(\mathbb{D}_{r_{1,2}}),f(A(r_{1,2},1)))$ is extremal for Problem $P_{2}$.
Also, conformal mappings preserve the module of a doubly connected domain.
Hence,

$m(f(A(r_{1,2},1)))=m(A(r_{1,2},1))=-\frac{1}{2\pi}\log r_{1,2}.$ (4)

Now, it follows from Theorem 2(3) that the free boundary of Problem $P_{2}$
lies in the annulus $\bar{A}(\rho_{1},\rho_{2})$. This implies that
$f(A(r_{1,2},1))\subset A(\rho_{1},1)$. Since the module of a doubly connected
domain increases under expansion, it follows that $m(f(A(r_{1,2},1)))\leq
m(A(\rho_{1},1))=-\frac{1}{2\pi}\log\rho_{1}$. Since $r_{1,2}<\rho_{1}$, the
latter inequality contradicts (4). Thus, our assumption leads to a
contradiction and therefore we must have $S_{\Omega}(0)=r_{1,2}$.

It remains to show that rotations about the origin are the only extremal
functions for the problem under consideration. If $f$ is not a rotation, then
our argument above implies that $f(A(r_{1,2},1))$ is a proper subdomain of
$A(r_{1,2},1)$. Therefore,
$m(f(A(r_{1,2},1)))<m(A(r_{1,2},1))=-\frac{1}{2\pi}\log r_{1,2}$, which again
contradicts equation (4). The proof is complete. $\Box$

More results and questions. Let $\Omega$ be a domain (of any connectivity) and
let $\gamma$ be a nondegenerate boundary continuum of $\Omega$ separated from
the rest of $\partial\Omega$, i.e. such that
$\gamma\cap(\overline{\partial\Omega\setminus\gamma})=\emptyset$. If $\Omega$
is finitely connected, the latter separation property always holds. Then, for
each $z\in\Omega$, we can consider a non-empty class
$\mathcal{U}_{\gamma}(\Omega)$ of functions $f\in\mathcal{U}(\Omega)$ such
that $f(\gamma)=\mathbb{T}$, in the sense of boundary correspondence. Then the
function $S_{\Omega,\gamma}:\Omega\to\mathbb{R}$ defined by

$S_{\Omega,\gamma}(z)=\sup\\{{\mbox{dist}}(0,\partial
f(\Omega)):\,f\in\mathcal{U}\gamma(\Omega),\,f(z)=0\\}$ (5)

can be thought as the squeezing function toward the boundary continuum
$\gamma$. For this function we have the following result.

######  Lemma 1.

Let $\Omega$ and $\gamma$ be as above and let $a\in\Omega$. Suppose that there
is a function $f_{a,\gamma}\in\mathcal{U}_{\gamma}(\Omega)$ such that
$f_{a,\gamma}(a)=0$ and $\mathbb{D}\setminus
f_{a,\gamma}(\Omega)\subset\\{z:\,|z|=r\\}$, $0<r<1$. Then
$S_{\Omega,\gamma}(a)=r$.

Furthermore, $f_{a,\gamma}$ is a unique (up to rotation about $0$) function in
$\mathcal{U}_{\gamma}(\Omega)$ that is extremal for the problem on the
squeezing toward $\gamma$ for the point $a$.

_Proof._ Since the squeezing problem is conformally invariant, we may assume
that $\Omega$ is the disk $D$ slit along a proper compact subset of the circle
$T_{r}=\\{z:\,|z|=r\\}$, $0<r<1$. Then our argument used in the proof of
Theorem 3 shows that if $f\in\mathcal{U}_{\mathbb{T}}(\Omega)$ such that
$f(0)=0$ is extremal for the squeezing toward $\mathbb{T}$ problem, then
$\mathbb{D}\setminus f(\Omega)\subset\mathbb{T}_{r}$. Furthermore, same
argument shows that $f(\mathbb{D}_{r})=\mathbb{D}_{r}$. Since $f(0)=0$, the
latter implies that $f$ is a rotation about the origin. $\Box$

Below, we assume that $\Omega$ is a finitely-connected domain with $n\geq 3$
nondegenerate boundary continua $\gamma_{1},\ldots,\gamma_{n}$. In this case,
the following properties are either exist in the literature or easy to prove:

  1. (1)

Existence of extremal functions. For each $z\in\Omega$ and $1\leq k\leq n$,
there is an extremal function for the problem on squeezing toward
$\gamma_{k}$; i.e. a function $f_{z,k}\in\mathcal{U}_{\gamma_{k}}(\Omega)$
such that $f_{z,k}(z)=0$ and
$S_{\Omega,\gamma_{k}}(z)={\mbox{dist}}(0,\partial f_{z,k}(\Omega))$.
Therefore, for each $z\in\Omega$, there is a function
$f_{\Omega}\in\mathcal{U}(\Omega)$ extremal for the squeezing problem on
$\Omega$.

  2. (2)

Continuity. The functions $S_{\Omega}(z)$ and $S_{\Omega,\gamma_{k}}(z)$,
$k=1,\ldots,n$, are continuous on $\Omega$.

  3. (3)

Monotonicity of $S_{\Omega,\gamma_{n}}(z)$ with respect to $\gamma_{n}$.  For
a fixed $z\in\Omega$, $S_{\Omega,\gamma_{n}}(z)$ is monotone with respect to
$\gamma_{n}$ in the following sense. Let $\Omega^{\prime}\not=\Omega$ be a
finitely connected domain with boundary continua
$\gamma^{\prime}_{1},\ldots,\gamma^{\prime}_{n^{\prime}}$, $2\leq
n^{\prime}\leq n$, such that $a\in\Omega^{\prime}\subset\Omega$ and
$\gamma^{\prime}_{k}=\gamma_{k}$ for $k=1,\ldots,n^{\prime}-1$. Then

$S_{\Omega^{\prime},\gamma^{\prime}_{n^{\prime}}}(a)>S_{\Omega,\gamma_{n}}(a).$

  4. (4)

Non-monotonicity of $S_{\Omega}(z)$ with respect to $\Omega$. For a fixed
$z\in\Omega$, $S_{\Omega}(z)$ is not monotone as a function of the domain.

  5. (5)

Boundary values of $S_{\Omega}(z)$. $S_{\Omega,\gamma_{k}}(z)\to 1$ as
$z\to\gamma_{k}$ and therefore $S_{\Omega}(z)\to 1$ as $z\to\partial\Omega$.

  6. (6)

Boundary values of $S_{\Omega,\gamma_{k}}(z)$ on $\gamma_{j}$, $j\not=k$. For
each $k$ and $j\not=k$, there is $0<c_{j,k}<1$ such that

$\limsup_{z\to\gamma_{j}}S_{\Omega,\gamma_{k}}(z)\leq c_{j,k}.$

  7. (7)

Separation property 1. If $c$, $0<c<1$, is sufficiently close to $1$, then the
level set $\\{z\in\Omega:\,S_{\Omega,\gamma_{k}}(z)=c\\}$ separates
$\gamma_{k}$ from $\partial\Omega\setminus\gamma_{k}$.

  8. (8)

Separation property 2. Let $j\not=k$. The set
$\\{z\in\Omega:\,S_{\Omega,\gamma_{j}}(z)=S_{\Omega,\gamma_{k}}(z)\\}$
separates $\gamma_{j}$ from $\gamma_{k}$ inside $\Omega$.

(1) The existence of extremal functions and therefore existence of extremal
domains as well was established in [1, Theorem 2.1].

(2) To prove the continuity property, one can use extremal functions
$f_{z,\Omega}(\zeta)$ composed with the Moebius automorphisms of the unit disk
$\mathbb{D}$. The details are left to the interested reader.

(3) To prove the monotonicity property of the squeezing function
$S_{\Omega,\gamma_{n}}(z)$ defined by (5), we consider the composition
$\varphi\circ f_{\Omega,\gamma_{n}}$ of the extremal function
$f_{\Omega,\gamma_{n}}$ with the function $\varphi$ that is the Riemann
mapping function from a simply connected domain with the boundary continuum
$\gamma^{\prime}_{n^{\prime}}$ such that $\varphi(0)=0$. Then the desired
result follows from the Schwarz’s lemma.

(4) To show that the monotonicity with respect to $\Omega$ is absent, consider
two simple examples. Let $\Omega$ be a circularly slit disk
$\mathbb{D}\setminus\\{re^{i\theta}:\,|\theta|\leq\alpha\\}$ with $0<r<1$,
$0<\alpha<\pi$. For $0<r_{1}<r$, $r<r_{2}<1$, let
$\Omega_{1}=\Omega\setminus[r_{1},r]$, $\Omega_{2}=\Omega\setminus[r,r_{2}]$.
Then the argument involving Theorem 2, as it was used in the proof of Theorem
1, shows that $S_{\Omega_{1}}(0)<S_{\Omega}(0)<S_{\Omega_{2}}(0)$. Therefore,
the monotonicity property of $S_{\Omega}(z)$ as a function of the domain does
not hold, in general.

(5) To show that $S_{\Omega,\gamma_{k}}(z)\to 1$ as $z\to\gamma_{k}$, we may
assume that $\Omega\subset\mathbb{D}$ and $\gamma_{k}=\mathbb{T}$. Then the
Moebius mapping $\varphi(\zeta)=(\zeta-z)/(1-\bar{z}\zeta)$ is in
$\mathcal{U}_{\gamma_{k}}(\Omega)$ and it is easy to see that that
${\mbox{dist}}(0,\partial\varphi(\Omega))\to 1$ as $|z|\to 1$.

(6) Let $l$ be a Jordan curve in $\Omega$ separating $\gamma_{j}$ from
$\partial\Omega\setminus\gamma_{j}$ and let $\Omega_{l}$ be a doubly connected
domain with boundary components $\gamma_{j}$ and $l$. It follows from the
property (3) that $S_{\Omega}(z)<S_{\Omega_{l}}(z)$ for all $z\in\Omega_{l}$.

Fix $z_{0}\in\Omega_{l}$. Let $m=m(D_{l})$ be the module of the doubly
connected domain $D_{l}$. Then there is a function $\varphi$ that maps
$\Omega_{l}$ conformally onto the annulus $A(s,1)$ with $s=e^{-2\pi m}$. In
addition, we may assume that $\varphi(z_{0})=a$, $s<a<1$. Furthermore, there
exists a function $\psi$, such that $\psi(a)=0$, which maps $A(s,1)$
conformally onto the disk $\mathbb{D}$ slit along and arc $C$ on a circle
$\mathbb{T}_{\rho}$, with $0<\rho<1$, where $\rho=\rho(a)$ depends on $a$. The
mapping $\psi$ is one of the well-studied canonical mappings. A particular
property of $\psi$, we need here, was proved by E. Reich and S. E. Warschawski
in 1960 [9, Lemma 3]. These authors showed that $\rho(a)=a$. This result
implies that $a\to s$ as $z_{0}\to\gamma_{j}$. It follows from Theorem 1 that
$S_{\Omega_{l}}(z_{0})=\rho(a)=a$. This, being combined with property (3),
implies that

$\limsup_{z_{0}\to\gamma_{j}}\leq\lim_{a\to s}a=s=e^{-2\pi m}<1,$

as required.

(7) Let $c_{k}=\max_{j\not=k}\\{c_{j,k}\\}$, where $c_{j,k}$ introduced in
part (6) above. Take $c$, $c_{k}<c<1$, and consider the level set
$L(c)=\\{z\in\Omega:S_{\Omega,\gamma_{k}}(z)=c\\}$. Since
$S_{\Omega,\gamma_{k}}$ is continuous on $\Omega$ and has boundary value $1$
on $\gamma_{k}$ and boundary values less than $c$ on other boundary continua,
it follows that $L(c)$ separates $\gamma_{k}$ from the rest of
$\partial\Omega$.

(8) The same argument, which was used in part (7), can be used to prove the
separation property in question as well.

Let $\Omega$ be an $n$-connected domain as above. Then, for each $z\in\Omega$,
there are $n$ functions $\varphi_{z,k}$, $1\leq k\leq n$, and for each $k$
there are $n-1$ radii $r_{j,k}=r_{j,k}(z)$ of circular arcs
$C_{j,k}=C_{j,k}(z)$, as described above. An example of the domain $\Omega$
and its image under one of the mappings $\varphi_{z,k}$ are shown in Figure 2.
It was conjectured in [8] that

$S_{\Omega}(z)=\max_{k}\min_{j\not=k}r_{j,k}(z).$ (6)

Figure 2. Mapping onto a circularly slit disk.

This conjecture, though it sounds plausible, was quickly disproved by Gumenyuk
and Roth [3], whose engineering construction shows that for every $n\geq 3$
there is an $n$-connected domain $\Omega$, that is not a circularly slit disk,
and there is a point $z\in\Omega$ such that $S_{\Omega}(z)$ is strictly
greater than the right-hand side of (6). This shows that, in general, the
family of domains extremal for the squeezing problem is not limited to the set
of circularly slit disks. In [8] and [3], the question whether or not there
exist circularly slit disks of connectivity $n\geq 3$ that are extremal for
the squeezing problem was left open. Thus, we complement results in [8] and
[3] with the following theorem.

######  Theorem 3.

For each $n\geq 3$, there is a domain $\Omega$, that is a circularly slit disk
of connectivity $n$, such that $\Omega$ is the extremal domain, unique up to
rotation about the origin, for the squeezing problem for the point $z=0$.

_Proof._ Let $n\geq 3$, $0<r<1$, and $0<\alpha<\pi/(n-1)$. Let
$\gamma_{k}=\\{z=re^{i\theta}:\,|\theta-2\pi(k-1)/(n-1)|\leq\alpha\\}$ for
$1\leq k\leq n-1$ and let $\gamma_{n}=\mathbb{T}$. Let
$\Omega=\Omega(n,r,\alpha)$ denote the disk $\mathbb{D}$ slit along the arcs
$\Gamma_{k}$, $k=1,\ldots,n-1$.

It follows from Lemma 1 that $S_{\Omega,\gamma_{n}}(0)=r$. Furthermore, since
$\Omega$ possesses $(n-1)$-fold rotational symmetry, it follows that there is
$\rho$, $0<\rho<1$, such that

$S_{\Omega,\gamma_{k}}(0)=\rho\quad\quad{\mbox{for $k=1,\ldots,n-1$.}}$

We claim that, for fixed $n$ and $r$, $\rho=\rho(\alpha)\to 0$ as $\alpha\to
0$. To emphasize dependence on $\alpha$, we use notations
$\Omega=\Omega(\alpha)$, $\gamma_{k}=\gamma_{k}(\alpha)$, etc. To prove the
claim, we consider a circle $\Gamma_{\varepsilon}=\\{z:\,|z-r|=\varepsilon\\}$
with $\varepsilon>0$ sufficiently small. If $\alpha$ is small, then
$\Gamma_{\varepsilon}$ separates $\gamma_{1}(\alpha)$ from all other boundary
components of $\Omega(\alpha)$. Let $D_{\varepsilon}(\alpha)$ denote the
doubly connected domain bounded by $\Gamma_{\varepsilon}$ and
$\gamma_{1}(\alpha)$. It is clear that $m(D_{\varepsilon}(\alpha))\to\infty$
as $\alpha\to 0$. Since the module of a doubly connected domain is conformally
invariant, it follows that for every $\delta>0$ there is $\alpha_{0}>0$ such
that $f(\Gamma_{\varepsilon})\subset\mathbb{D}_{\delta}$, whenever
$0<\alpha<\alpha_{0}$ and
$f\in\mathcal{U}_{\gamma_{1}(\alpha)}(\Omega(\alpha))$ is such that $f(0)=0$.
Since the images $f(\mathbb{T})$ and $f(\gamma_{k}(\alpha))$,
$k=2,\ldots,n-1$, lie in the bounded component of $\mathbb{C}\setminus
f(\Gamma_{\varepsilon})$, the claim follows. Therefore, $\rho(\alpha)<r$ if
$\alpha$ is small enough. Hence $S_{\Omega(\alpha)}(0)=r$ for all such
$\alpha$ and $\Omega(\alpha)$ is the only extremal domain up to rotation about
the origin. The proof is complete. $\Box$

Example. Given $a>1$, let $\gamma_{k}=\gamma_{k}(a)=\\{z=te^{2\pi
i(k-1)/3}:\,1\leq t\leq a\\}$, $k=1,2,3$. Then
$\Omega=\overline{\mathbb{C}}\setminus\cup_{k=1}^{3}\gamma_{k}$ is a domain on
$\overline{\mathbb{C}}$ of connectivity $3$. Let $\varphi_{0,1}$ map $\Omega$
conformally onto a circularly slit disk $\mathbb{D}$ such that
$\varphi_{0,1}(\gamma_{1})=\mathbb{T}$, $\varphi_{0,1}(0)=0$, and
$\varphi^{\prime}_{0,1}(0)>0$. Since $\Omega$ is invariant under rotation by
angle $2\pi/3$ about $0$, the symmetry principle implies that
$\varphi_{0,1}(\gamma_{2})$ and $\varphi_{0,1}(\gamma_{3})$ are circular arcs
lying on the same circle symmetrically to each other with respect to the real
axis. Therefore, it follows from Theorem 3 that $\varphi_{0,1}$ is extremal
for the squeezing toward $\gamma_{1}$ problem for the point $0$. Since
$\Omega$ is invariant under rotations by the angle $2\pi/3$, it follows that
each of the other two functions, $\varphi_{0,2}$ and $\varphi_{0,3}$, where
$\varphi_{0,k}$ maps $\gamma_{k}$ to the unit circle $\mathbb{T}$ such that
$\varphi_{0,k}(0)=0$, $k=2,3$, is also extremal for the squeezing problem
toward, respectively, $\gamma_{2}$ or $\gamma_{3}$, for the point $0$.
Moreover,
$S_{\Omega,\gamma_{1}}(0)=S_{\Omega,\gamma_{2}}(0)=S_{\Omega,\gamma_{3}}(0)$.

Thus, this example shows that there are domains $\Omega$ of connectivity $n>2$
and points $z_{0}\in\Omega$, such that every function $f$ extremal for the
squeezing problem for $\Omega$ and $z_{0}$ maps $\Omega$ onto a circularly
slit disk.

Questions. We finish this part with six questions for future study.

  1. (1)

As Theorem 1 shows, if $\Omega$ is doubly connected, then every function $f$
extremal for the squeezing problem for some point $z_{0}\in\Omega$ maps
$\Omega$ onto a circularly slit disk. We suspect that this property
characterizes doubly connected domains. So, the question: Is it true that if
there is an open subset $G$ of $\Omega$, such that any function $f$ extremal
for the squeezing problem for some $z\in G$ maps $\Omega$ onto a circularly
slit disk, then $\Omega$ is doubly connected?

  2. (2)

A point $z_{0}$ in a domain $\Omega$ with boundary continua
$\gamma_{1},\ldots,\gamma_{n}$, $n\geq 2$, is an equilibrium point for the
squeezing problem if
$S_{\Omega,\gamma_{1}}(z_{0})=S_{\Omega,\gamma_{2}}(z_{0})=\ldots=S_{\Omega,\gamma_{n}}(z_{0})$.
An annulus $A(r,1)$ has the circle $\\{z:\,|z|=\sqrt{r}\\}$ as its set of
equilibrium points. A domain
$D_{n}(a)=\overline{\mathbb{C}}\setminus\cup_{k=1}^{n}\\{te^{2\pi
i(k-1)/n}:\,1\leq t\leq a\\}$ with $a>1$ and $n\geq 3$ has two equilibrium
points, $z_{1}=0$ and $z_{2}=\infty$. Let
$D_{n}(a,R)=D_{n}(a)\cap\mathbb{D}_{R}$, where $R>a$. There is a unique value
$R_{0}>a$ such that $D_{n}(a,R)$ has exactly one equilibrium point $z_{0}=0$,
while, for $R\not=R_{0}$, $D_{n}(a,R)$ does not have equilibrium points.

Is it true that every domain $\Omega$ of connectivity $n\geq 3$ has at most
two equilibrium points?

  3. (3)

Are there circularly slit disks with boundary continua $\gamma_{n}=\mathbb{T}$
and $\gamma_{k}\subset\mathbb{D}$, $k=1,\ldots,n$, $n\geq 3$, that are
extremal for the squeezing problem for $z=0$, such that $\gamma_{j}$ and
$\gamma_{k}$ lie on different circles when $j\not=k$?

  4. (4)

Are there domains $\Omega\subset\mathbb{D}$ of connectivity $n+m+1$ with
boundary continua $\gamma_{n+m+1}=\mathbb{T}$ and
$\gamma_{k}\subset\mathbb{D}$, $k=1,\ldots,n+m$, such that $\gamma_{k}$ is an
arc of a circle centered at $0$ when $1\leq k\leq n$, $\gamma_{k}$ is not a
circular arc when $n+1\leq k\leq n+m$, that are extremal for the squeezing
problem for $z=0$?

  5. (5)

Is it possible to characterize domains $\Omega$ of connectivity $n\geq 3$,
that are extremal for the squeezing problem for a point $z_{0}\in\Omega$, in
terms of quadratic differentials similar to characterization of configurations
of domains extremal for Jenkins’s problem [5] on extremal partitioning?

  6. (6)

Our proof of Theorem 2 on the extremal partitioning of the disk $\mathbb{D}$
with a compact barrier $E\subset\mathbb{D}$ can be extended to higher
dimensions for the unit ball $\mathbb{B}\subset\mathbb{R}^{n}$, $n\geq 3$,
with a compact set $E\subset\mathbb{B}$.

It would be interesting to know whether such a generalization can be used to
study the squeezing problem in $\mathbb{C}^{n}$.

Proof of Theorem 2. Problem M is a particular case of Jenkins’s problem on
extremal partitioning [5]. Therefore, parts (1) and (2) of Theorem 2 follow
from [5, Theorem 1]. Furthermore, same Jenkins’s theorem implies that the
metric $\rho_{1}(z)|dz|=\alpha_{1}^{-1}|Q(z)|^{1/2}\,|dz|$ is extremal for the
module problem for the family of closed curves $\gamma\subset D_{1}^{*}$
separating $z=0$ from $\partial D_{1}^{*}$ and the metric
$\rho_{2}(z)|dz|=\alpha_{2}^{-1}|Q(z)|^{1/2}\,|dz|$ is extremal for the module
problem for the family of closed curves $\gamma\subset D_{2}^{*}$ separating
$\mathbb{T}$ from $\partial D_{2}^{*}\setminus\mathbb{T}$. Moreover, it
implies that

$\int_{\gamma}|Q(z)|^{1,2}\,|dz|\geq\alpha_{2}$ (7)

for every Jordan curve $\gamma\subset\mathbb{D}\setminus E$, which separates
$\mathbb{T}$ from $E$ and $0$ and also for $\gamma=\mathbb{T}$.

Let us we consider the case $\alpha_{1}\leq\alpha_{2}$, assuming that $E$
consists of a finite number of connected components. In this case
$D_{2}^{*}\not=\emptyset$. Otherwise, $\mathbb{T}$ would be on the boundary of
$D_{1}^{*}$, which implies that

$\int_{\mathbb{T}}|Q(z)|^{1/2}\,|dz|<\alpha_{1}\leq\alpha_{2},$

contradicting equation (7). Furthermore, since $E$ consists of a finite number
of components, it follows that the free boundary $L(E)$ consists of a finite
number of critical trajectories of $Q(z)\,dz^{2}$ and their endpoints in
$\mathbb{D}\setminus E$.

Suppose by contradiction that

$r_{2}<\rho=\max\\{|z|:\,z\in L(E)\\}<1.$ (8)

Since $L(E)$ consists of a finite number of analytic arcs, it follows that the
intersection $L(E)\cap\mathbb{T}_{\rho}$ consists of a finite number of points
$z_{k}$, $k=1,\ldots,n$. It follows from the local structure of trajectories
of $Q(z)\,dz^{2}$ near critical points, that each of the points $z_{k}$ is
regular. This implies that, for all sufficiently small $\varepsilon>0$, the
set $L(E)\cap\bar{A}(\rho-\varepsilon,\rho)$ consists of $n$ disjoint analytic
arcs $s_{k}$, $k=1,\ldots,n$, such that $s_{k}$ has its endpoints on the
circle $\mathbb{T}_{\rho-\varepsilon}$ and the point $z_{k}$ is an interior
point of $s_{k}$. Let $\tilde{s}_{k}$ denote the arc symmetric to $s_{k}$ with
respect to the circle $\mathbb{T}_{\rho-\varepsilon}$. For every $\varepsilon$
small enough and each $k$, the intersection $\tilde{s}_{k}\cap L(E)$ is empty.
In this case, there are $n$ simply connected domains $\Delta_{k}$, symmetric
with respect to the circle $\mathbb{T}_{\rho-\varepsilon}$, such that
$\partial\Delta_{k}=s_{k}\cup\tilde{s}_{k}$, $k=1,\ldots,n$.

We claim that there is no $k$ such that $\Delta_{k}\subset D_{2}^{*}$. To
prove this claim, suppose that $\Delta_{k_{j}}\subset D_{2}^{*}$ for
$j=1,\ldots,n_{1}$, $n_{1}\leq n$. Let $\partial^{0}D_{2}^{*}=\mathbb{T}$ and
$\partial^{1}D_{2}^{*}$ denote the boundary components of $D_{2}^{*}$. Let
$D_{2}^{p}$ denote polarization of the doubly connected domain $D_{2}^{*}$
(considered as a condenser whose plates are connected components of
$\mathbb{C}\setminus D_{2}^{*}$) with respect to the circle
$\mathbb{T}_{\rho-\varepsilon}$. In the case under consideration, the
polarized domain $D_{2}^{p}$ is a doubly connected domain having $\mathbb{T}$
as one of its boundary components while the other boundary component of
$D_{2}^{p}$ is obtained from $\partial^{1}D_{2}^{*}$ by replacing the arcs
$s_{k_{j}}$ with the arcs $\tilde{s}_{k_{j}}$, $j=1,\ldots,n_{1}$. For the
definition and properties of polarization, we refer to [2, Chapter 3]. As well
known, polarization increases the module of a doubly connected domain. Thus,
in our case, we have

$m(D_{2}^{*})<m(D_{2}^{p}),$

with the sign of strict inequality because $D_{2}^{p}$ does not coincide with
$D_{2}^{*}$ up to reflection with respect to $\mathbb{T}_{\rho-\varepsilon}$.
Therefore,

$\alpha_{1}^{2}m(D_{1}^{*},0)+\alpha_{2}^{2}m(D_{2}^{*})<\alpha_{1}^{2}m(D_{1}^{*},0)+\alpha_{2}^{2}m(D_{2}^{p}).$

Since the pair $(D_{1}^{*},D_{2}^{p})$ is admissible for Problem M, i.e.
$(D_{1}^{*},D_{2}^{p})\in\mathcal{D}(E)$, inequality (8) contradicts our
assumption that the pair of domains $(D_{1}^{*},D_{2}^{*})$ is extremal for
Problem M.

Now we consider the case when $\Delta_{k}\subset D_{1}^{*}$ for all
$k=1,\ldots,n$. In this case, we consider a pair $(D_{1},D_{2})$, where
$D_{1}=D_{1}^{*}\setminus\bar{A}(\rho-\varepsilon,\rho)$ is a simply connected
domain and $D_{2}=D_{2}^{*}\cup A(\rho-\varepsilon,\rho)\cup\mathbb{T}_{\rho}$
is a doubly connected domain. One can easily see that
$(D_{1},D_{2})\in\mathcal{D}(E)$. Let $\rho_{k}(z)|dz|$ denote the extremal
metric for the corresponding module problem for the domain $D_{k}$, $k=1,2$;
see [4, Chapter II].

Since $D_{2}^{*}\subset D_{2}$, it follows that $\rho_{2}(z)|dz|$ is
admissible for the module problem for $D_{2}^{*}$ but it is not extremal for
this problem. Therefore,

$\displaystyle m(D_{2}^{*})$
$\displaystyle<\iint_{D_{2}}\rho_{2}^{2}(z)\,dA-\iint_{\cup_{k=1}^{N}\Delta_{k}}\rho_{2}^{2}(z)\,dA$
(9)
$\displaystyle=m(D_{2})-\iint_{\cup_{k=1}^{N}\Delta_{k}}\rho_{2}^{2}(z)\,dA.$

Next, we consider a metric $\rho(z)|dz|$ on the domain $D_{1}^{*}$ defined by

$\rho(z)=\begin{cases}\rho_{1}(z)&{\mbox{if $z\in D_{1}$,}}\\\
\rho_{2}(z)&{\mbox{if $z\in D_{1}^{*}\setminus D_{1}$.}}\end{cases}$

We claim that $\rho(z)|dz|$ is admissible for the problem on the reduced
module for the domain $D_{1}^{*}$. To prove this, we consider an analytic
Jordan curve $\gamma\subset D_{1}^{*}$ separating $0$ from $\partial
D_{1}^{*}$. If $\gamma\subset\bar{D}_{1}$, then

$\int_{\gamma}\rho(z)\,|dz|=\int_{\gamma}\rho_{1}(z)\,|dz|\geq 1$ (10)

since $\rho_{1}(z)|dz|$ is admissible for the problem on the reduced module of
$D_{1}$.

Suppose now that $\gamma\not\subset\bar{D}_{1}$. Then the circle
$\mathbb{T}_{\rho-\varepsilon}$ divides $\gamma$ into a finite number of arcs.
By $\tau_{k}$, $k=1,\ldots,m$, we denote those of them, which lie in $D_{2}$.
Let $\sigma_{k}\subset D_{1}^{*}$ denote the closed arc of
$T_{\rho-\varepsilon}$ joining the endpoints of $\tau_{k}$, $k=1,\ldots,m$. It
follows from Lemma 2, presented below, that

$\int_{\tau_{k}}\rho(z)\,|dz|\geq\int_{\sigma_{k}}\rho(z)\,|dz|\geq\int_{\sigma_{k}}\rho_{1}(z)\,|dz|.$
(11)

Let $\tilde{\gamma}$ denote the curve obtained from $\gamma$ by replacing the
arcs $\tau_{k}$ with the arcs $\sigma_{k}$, $k=1,\ldots,m$. Then
$\tilde{\gamma}$ is the curve in $\bar{D}_{1}$, that is not Jordan in general,
that separates $0$ from $\partial D_{1}$. Hence,

$\int_{\tilde{\gamma}}\rho_{1}(z)\,|dz|\geq 1.$ (12)

Combining equations (11) and (12), we conclude that

$\int_{\gamma}\rho(z)\,|dz|\geq\int_{\tilde{\gamma}}\rho_{1}(z)\,|dz|\geq 1.$
(13)

Equations (10) and (13) imply that $\rho(z)|dz|$ is an admissible metric for
the problem on the reduced module in $D_{1}^{*}$. Hence,

$\displaystyle m(D_{1}^{*},0)$ $\displaystyle\leq\lim_{\epsilon\to
0}\left\\{\iint_{D_{1}^{*}\setminus\bar{\mathbb{D}}_{\epsilon}}\rho^{2}(z)\,dA+\frac{1}{2\pi}\log\epsilon\right\\}$
(14) $\displaystyle=\lim_{\epsilon\to
0}\left\\{\iint_{D_{1}\setminus\bar{\mathbb{D}}_{\epsilon}}\rho^{2}(z)\,dA+\iint_{\cup_{k=1}^{N}\Delta_{k}}\rho_{2}^{2}(z)\,dA+\frac{1}{2\pi}\log\epsilon\right\\}$
$\displaystyle=m(D_{1},0)+\iint_{\cup_{k=1}^{N}\Delta_{k}}\rho_{2}^{2}(z)\,dA.$

Combining (9) and (14), we obtain the following inequalities:

$\displaystyle\alpha_{1}^{2}m(D_{1}^{*},0)$
$\displaystyle+\alpha_{2}^{2}m(D_{2}^{*})<\alpha_{1}^{2}m(D_{1},0)+\alpha_{2}^{2}m(D_{2})$
$\displaystyle+(\alpha_{2}^{2}-\alpha_{1}^{2})\iint_{\cup_{k=1}^{N}\Delta_{k}}\rho_{2}^{2}(z)\,dA\leq\alpha_{1}^{2}m(D_{1},0)+\alpha_{2}^{2}m(D_{2}).$

The latter inequalities contradict to the assumption that the pair
$(D_{1}^{*},D_{2}^{*})$ is extremal for Problem M. This contradiction shows
that our assumption $R>r_{2}$ is wrong and therefore,
$L(E)\subset\bar{\mathbb{D}}_{r_{2}}$, as required.

The latter inclusion is proved in the case when $E$ consists of a finite
number of components. In the general case, we approximate $E$ with a sequence
of compact sets $E^{j}$, $j=1,2,\ldots$, such that $E^{j+1}\subset E^{j}$ and
such that $E^{j}$ is bounded by appropriate level curves of Green’s function
$g_{\mathbb{D}\setminus E}(z,0)$ of the domain $\mathbb{D}\setminus E$ with
pole at $0$. Then the required inclusion $L(E)\subset\bar{\mathbb{D}}_{r_{2}}$
will follow from the result already proved for sets $E$ with finite number of
connected components and from Carathéodory’s convergence theorem for simply
connected and doubly connected domains.

The proof presented above, can be easily modified to show that if
$\alpha_{1}\geq\alpha_{2}$, then $L(E)\subset\bar{A}(r_{1},1)$. $\Box$

######  Lemma 2.

Let $\sigma$ be an open arc on $\mathbb{T}$. Let $D_{k}$, $k=1,2$, be a doubly
connected domain having the circle $\mathbb{T}_{r}$ , $0<r<1$, as one of its
boundary components. Suppose that the other boundary component of $D_{1}$,
call it $\gamma_{1}$, is such that $l\subset\gamma_{1}$ and $D_{1}\subset
A(r,1)$. Suppose further that the other boundary component of $D_{2}$, call it
$\gamma_{2}$, is such that $l\subset\gamma_{2}$ and $A(r,1)\subset D_{2}$. Let
$\rho_{k}(z)|dz|$ denote the extremal metric of the module problem in $D_{k}$,
$k=1,2$. Then

$\rho_{1}(e^{i\theta})\leq 1/2\pi\leq\rho_{2}(e^{i\theta})\quad{\mbox{for all
$e^{i\theta}\in\sigma$.}}$ (15)

_Proof._ Let us prove the first inequality. Let $f_{1}$ maps $D_{1}$
conformally onto the annulus $A(r_{1},1)$ such that
$f_{1}(\gamma_{1})=\mathbb{T}$. Since the module of a doubly connected domain
increases under expansion, the following holds: $r<r_{1}<1$. In terms of the
mapping function, the extremal metric can be expressed as follows (see [4,
Chapter II]):

$\rho_{1}(z)|dz|=\frac{1}{2\pi}\frac{|f^{\prime}_{1}(z)|}{|f(z)|}\,|dz|.$

This shows that the desired result will follow if we prove that

${\mbox{meas}}(f_{1}(\sigma))\leq{\mbox{meas}}(\sigma)$ (16)

for every $\sigma\subset\mathbb{T}$, if the assumptions of the lemma are
satisfied.

To prove (16), consider a family of curves $\Gamma_{1}(\sigma)=\\{\gamma\\}$
consisting of all rectifiable arcs $\gamma$ joining $\mathbb{T}_{r}$ and
$\sigma$ inside the domain $D_{1}$. Let $\Gamma(s,\sigma)$ denote a similar
family of curves in the annulus $A(s,1)$, $0<s<1$. Since the module increases
under expansion of the family of curves, we have

${\mbox{mod}}(\Gamma_{1}(\sigma))<{\mbox{mod}}(\Gamma(s,\sigma))\quad{\mbox{if
$D_{1}\not=A(r,1)$.}}$ (17)

Now, suppose by contradiction that
${\mbox{meas}}(f_{1}(\sigma))>{\mbox{meas}}(\sigma)$. Since the module of a
family of curves is conformally invariant and since a family of shorter curves
has bigger module than a corresponding family of longer curves, we have the
following:

${\mbox{mod}}(\Gamma_{1}(\sigma))={\mbox{mod}}(\Gamma(r_{1},f_{1}(\sigma)))>{\mbox{mod}}(\Gamma(r,f_{1}(\sigma)))\geq{\mbox{mod}}(\Gamma(s,\sigma)),$
(18)

contradicting (17). Thus, the inequality (16) is proved and therefore the
first inequality in (15) holds. The proof of the second inequality in (15)
follows the same lines. $\Box$

Notice that (15) holds for doubly connected domains $D_{k}$ having the circle
$\mathbb{T}_{r}$ of arbitrary small radius $0<r<1$ as one of its boundary
components. Therefore, taking the limit as $r\to 0$, we conclude that (15)
remains valid if $D_{1}$ and/or $D_{2}$ is a simply connected domain
containing $0$. In the latter case, $\rho_{k}(z)|dz|$ will denote the extremal
metric for the problem on the reduced module $m(D_{k},0)$.

In the case of simply connected domains, (15) also follows from the Loewner’s
lemma or from Carleman’s expansion principle for the harmonic measure.

Remarks. (1) As Gumenyuk and Roth showed in [3], Theorem 1 on the extremality
of circularly slit disks cannot be extended to domains of connectivity greater
than two. This result sounds similar to the observation made in [10] about a
possibility to extend Theorem 2 stated above in this note to the case of
several barriers $E_{k}$ contained in non-overlapping annuli
$\bar{A}(r_{k}^{1},r_{k}^{2})$, $0<r_{1}^{1}\leq r_{1}^{2}<r_{2}^{1}\leq
r_{2}^{2}<\cdots<r_{n}^{1}\leq r_{n}^{2}<1$; i.e. such that
$E_{k}\subset\bar{A}(r_{k}^{1},r_{k}^{2})$. Precisely, if
$(D_{1}^{*},\ldots,D_{n}^{*})$ is the configuration of non-overlapping domains
maximizing the weighted sum of moduli

$m(D_{1},0)+\sum_{k=2}^{n}m(D_{k}),$

then the free boundary $L(E_{1},\ldots,E_{n})=\cup_{k=1}^{n}\partial
D_{k}^{*}\cap(\mathbb{D}\setminus\cup_{k=1}^{n}E_{k})$ is not necessarily
contained in the union $\cup_{k=1}^{n}\bar{A}(r_{k}^{1},r_{k}^{2})$ of these
annuli. Here, $D_{1}\subset\mathbb{D}\setminus\cup_{k=1}^{n}E_{k}$ is a simply
connected domain containing $0$ and $D_{k}$, $k=2,\ldots,n$, is a doubly
connected domain in $\mathbb{D}\setminus\cup_{k=1}^{n}E_{k}$ separating the
set $\cup_{j=1}^{k-1}E_{j}$ from the unit circle $\mathbb{T}$ and the set
$\cup_{j=k}^{n}E_{j}$.

(2) Our proof of Theorem 2 relies on the technique, which uses weighted sums
of moduli of free families of curves developed by J. A. Jenkins and others. As
well known, a module of a doubly connected domain $D$ with complementary
components $E_{0}$ and $E_{1}$ is the reciprocal of the capacity, when $D$ is
considered as the field of the condenser with plates $E_{0}$ and $E_{1}$. This
observation suggests that an approach utilizing properties of capacities and
potential functions of condensers also can be used to study the squeezing
problem. An example, demonstrating how this approach works, was given in the
second proof of Theorem 1.3 in [10]. The proof of Theorem 2 in [3] is a nice
demonstration how this approach based on the potential theory can be used in
the context of the squeezing problem.

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