# Temperature on rods with Robin boundary conditions

Dimitrios Betsakos and Alexander Solynin

###### Abstract.

We consider solutions $u_{f}$ to the one-dimensional Robin problem with the
heat source $f\in L^{1}[-\pi,\pi]$ and Robin parameter $\alpha>0$. For given
$m$, $M$, and $s$, $0\leq m<s<M$, we identify the heat sources $f_{0}$, such
that $u_{f_{0}}$ maximizes the temperature gap
$\max_{[-\pi,\pi]}u_{f}-\min_{[-\pi,\pi]}u_{f}$ over all heat sources $f$ such
that $m\leq f\leq M$ and $\|f\|_{L^{1}}=2\pi s$. In particular, this answers a
question raised by J. J. Langford and P. McDonald in [5]. We also identify
heat sources, which maximize/minimize $u_{f}$ at a given point
$x_{0}\in[-\pi,\pi]$ over the same class of heat sources as above and discuss
a few related questions.

###### Key words and phrases:

Heating problem, Poisson equation, Robin boundary conditions, comparison
theorem

###### 2010 Mathematics Subject Classification:

34B08, 34C10

##  1\. Heating with Robin boundary conditions.

Recently, J. J. Langford and P. McDonald [5] studied the one-dimensional
Poisson equation with Robin boundary conditions. They considered the following
physical setup: Suppose that a metal rod of length $2\pi$ is located along the
interval $[-\pi,\pi]$. Suppose that to half of the locations of the rod, heat
is generated uniformly; call this set $E$. On the remaining half, heat is
neither generated nor absorbed. The ends of the rod interact with the cooler
environment so that there is a heat flux from the rod which is proportional to
the temperature at each end (Newton’s law of cooling).

Let $u$ be the steady-state temperature function; it satisfies the Poisson
equation

$-u^{\prime\prime}(x)=\chi_{E}(x),\;\;\;x\in[-\pi,\pi]$ (1.1)

with Robin boundary conditions

$-u^{\prime}(-\pi)+\alpha u(-\pi)=u^{\prime}(\pi)+\alpha u(\pi)=0,$ (1.2)

where $\alpha>0$ and $\chi_{E}$ stands for the characteristic function of the
set $E$.

Langford and McDonald [5] studied the problem of where one should locate the
heat sources to maximize the hottest steady-state temperature. In other words,
for which set $E$, for the solution $u$ of the boundary value problem
(1.1)-(1.2), $\max_{[-\pi,\pi]}u$ is maximal. They showed that this quantity
is maximal when $E$ is an interval located symmetrically in the middle of the
rod; namely $E=[-\pi/2,\pi/2]$. The authors of [5] actually studied a more
general problem and obtained much stronger comparison results. They observed,
however, that the symmetric interval is not extremal for another problem. They
considered the temperature gap over $[-\pi,\pi]$, i.e. the quantity

${\rm osc}(u):=\max_{[-\pi,\pi]}u-\min_{[-\pi,\pi]}u$

and showed that ${\rm osc}(u)$ is not maximized for $E=[-\pi/2,\pi/2]$. So
they raised the following question:

######  Problem 1.

Where should we place the heat sources to maximize the temperature gap?

They suggested that the extremal set $E$ is again an interval which, however,
is not symmetrically located on $[-\pi,\pi]$. In the present note, we will
study this conjecture. As in [5], we will consider a more general setting.

The Robin problem for $f\in L^{1}[-\pi,\pi]$ and $\alpha>0$ is to find $u\in
C^{1}[-\pi,\pi]$ such that

  1. 1.

$u^{\prime}$ is absolutely continuous on $[-\pi,\pi]$,

  2. 2.

$-u^{\prime\prime}=f$ a.e. on $(-\pi,\pi)$,

  3. 3.

$-u^{\prime}(-\pi)+\alpha u(-\pi)=u^{\prime}(\pi)+\alpha u(\pi)=0$.

It was shown in Proposition 2.1 in [5] that the Robin problem has a unique
solution given by the equation

$u_{f}(x)=\int_{-\pi}^{\pi}G(x,y)f(y)\,dy.$ (1.3)

Here, $G(x,y)$ stands for the Green’s function for Robin problem, which is

$G(x,y)=-\frac{1}{2}c_{\alpha}xy-\frac{1}{2}|x-y|+\frac{1}{2c_{\alpha}},\quad
x,y\in[-\pi,\pi],$ (1.4)

where

$c_{\alpha}=\frac{\alpha}{1+\alpha\pi}.$ (1.5)

To recall a few basic facts about solutions of the Robin problem, we note
first that, as simple Calculus shows, $G(x,y)>0$ for all $x,y\in[-\pi,\pi]$.
Another simple but important conclusion from (1.3) is that the solution to the
Robin problem is an additive function of the heat source; i.e. if $m_{1}$,
$m_{2}$ are constants and $f_{1},f_{2}\in L^{1}$ (here and below $L^{1}$
stands for $L^{1}[-\pi,\pi]$), then

$u_{f}=m_{1}u_{f_{1}}+m_{2}u_{f_{2}},\quad{\mbox{where
$f=m_{1}f_{1}+m_{2}f_{2}$.}}$ (1.6)

Furthermore, if $f\geq 0$, then it is immediate from property 2 above that
$u_{f}$ is concave on $[-\pi,\pi]$ and therefore it takes its minimal value at
one of the end points $x=\pm\pi$ and it takes its maximal value either at a
single point or on some closed subinterval of $[-\pi,\pi]$. If $u_{f}$ takes
its minimal value at $-\pi$, it follows from property 3 that $u_{f}(x)\geq
u_{f}(-\pi)=\alpha^{-1}u^{\prime}_{f}(-\pi)>0$ and therefore, in this case,
$u_{f}(x)>0$ for $x\in[-\pi,\pi]$ unless $f\equiv 0$. The same conclusion
follows if $u_{f}$ takes its minimal value at $\pi$.

To put the question in Problem 1 in a more general setting, we consider, for
given $m$, $M$, and $s$ such that $0\leq m<s<M$, the class
$\mathcal{F}=\mathcal{F}(m,M,s)$ of heat sources $f\in L^{1}$ such that $m\leq
f(x)\leq M$ for $x\in[-\pi,\pi]$ and $\|f\|_{L^{1}}=2\pi s$. The parameters
$m$, $M$, and $s$ can be interpreted as the ground heat, the top heat, and the
average heat over the rod.

Our main goal in this note is to prove the following theorem, which solves the
maximal temperature gap problem for the class $\mathcal{F}(m,M,s)$ and
therefore, as a special case, it provides a solution to Problem 1. In this
theorem and below, $I(a,l)\subset\mathbb{R}$ denotes the closed interval of
length $2l>0$ centered at $a$.

######  Theorem 1.

Let $u_{f}$ solves the Robin problem for $f\in\mathcal{F}(m,M,s)$ and
$\alpha>0$. Then

${\rm osc}(u_{f})\leq(M-m)\Theta_{\alpha}(l,\delta),$ (1.7)

where $l=\pi(s-m)/(M-m)$, $\delta=m/(M-m)$ and $\Theta_{\alpha}(l,\delta)$ is
defined by equations (4.11), (4.12) and (4.13) in Section 4.

Equality holds in (1.7) if and only if $f(x)=f_{0}(x)$ or $f(x)=f_{0}(-x)$
a.e. on $[-\pi,\pi]$, where $f_{0}=m+(M-m)\,\chi_{I(a_{g},l)}$ with
$l=\pi(s-m)/(M-m)$ and $a_{g}$ defined by equation (4.10) in Section 4.

For $m=1$, $M=3$, and $s=\frac{7}{5}$, the graph of the maximal temperature
gap $(M-m)\Theta_{\alpha}(l,\delta)$ considered as a function of $\alpha$ is
shown in Figure 1(a). Figure 1(b) displays the graph of the extremal function
$u_{f_{0}}(x)$, where $f_{0}=m+(M-m)\,\chi_{I(a_{g},l)}$ with $m=1$, $M=3$,
$s=\frac{7}{5}$ and $\alpha=1/2$.

(a)

(b)

Fig 1. (a) The maximal temperature gap $(M-m)\Theta_{\alpha}(l,\delta)$ for
$m=1$, $M=3$, $s=\frac{7}{5}$ and $0<\alpha<2$; (b) Extremal function
$u_{f_{0}}(x)$ for $m=1$, $M=3$, $s=\frac{7}{5}$ and $\alpha=1/2$.

Returning to the context of Problem 1, let us suppose that $E$ is a measurable
subset of $[-\pi,\pi]$ of length (one-dimensinal Lebesgue measure) equal to
$\pi$. We apply Theorem 1 with $f=\chi_{E}$, $m=0$, $M=1$, $\delta=0$, and
$l=\pi/2$. The parameter $\alpha_{0}$ defined by equation (4.8) in Section 4
takes the value $\alpha_{0}=\frac{2}{\sqrt{3}\pi}$. Let

$E^{*}=I(a_{g},\pi/2)=[a_{g}-\pi/2,a_{g}+\pi/2]\;\;\;\;\hbox{and}\;\;\;\;f^{*}=\chi_{E^{*}},$

where (see formula (4.10))

$a_{g}=\begin{cases}\pi/2,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\hbox{if}\;\;0<\alpha\leq\frac{2}{\sqrt{3}\pi},\\\
\frac{\pi/2}{(1+\alpha\pi)(\pi
c_{\alpha}/2-\pi^{2}c_{\alpha}^{2}/4)},\;\;\hbox{if}\;\;\alpha\geq\frac{2}{\sqrt{3}\pi}.\end{cases}$

It follows (cf. [5, Proposition 3.3]) that when
$0<\alpha\leq\frac{2}{\sqrt{3}\pi}$, we have $E^{*}=[0,\pi]$; when
$\alpha\geq\frac{2}{\sqrt{3}\pi}$, the location of the interval $E^{*}$
depends on $\alpha$ and as $\alpha$ increases, $E^{*}$ moves from the right
end to the center. By Theorem 1, we have ${\rm osc}(u_{f})\leq{\rm
osc}(u_{f^{*}})$. The solution to Problem 1 is: The temperature gap is
maximized uniquely when the heat sources are placed on $E^{*}$ or on $-E^{*}$.

Another interesting problem on the distribution of heat on a rod is to
identify heat sources $f\in\mathcal{F}(m,M,s)$, which generate the maximal
possible temperature and the minimal possible temperature at a fixed location
$x_{0}\in[-\pi,\pi]$ of the rod. Notice that if $f^{-}(x)=f(-x)$, then
$u_{f^{-}}(x)=u_{f}(-x)$. Thus, working with this problem, we may assume that
$x_{0}\in[0,\pi]$. Its solution is given by the following theorem.

######  Theorem 2.

Let $u_{f}$ solves the Robin problem for $f\in\mathcal{F}(m,M,s)$ and
$\alpha>0$ and let $x_{0}\in[0,\pi]$ be fixed. Then

$\eta_{\alpha}(x_{0})M-(M-m)\nu_{\alpha}(x_{0},l^{-})\leq
u_{f}(x_{0})\leq\eta_{\alpha}(x_{0})m+(M-m)\nu_{\alpha}(x_{0},l),$ (1.8)

where $l=\pi(s-m)/(M-m)$, $l^{-}=\pi-l$, and the functions $\eta_{\alpha}(x)$
and $\nu_{\alpha}(x,l)$ are defined in Section 2 by equations (2.5) and (2.7),
respectively.

Equality holds in the right inequality in (1.8) if and only if $f=f_{0}^{+}$
a.e. on $[-\pi,\pi]$, where $f_{0}^{+}=m+(M-m)\,\chi_{I(a_{m},l)}$ with $l$
defined above and $a_{m}=x_{0}(1-lc_{\alpha})$ if $x_{0}(1-lc\alpha)<\pi-l$
and $a_{m}=\pi-l$ otherwise.

Equality holds in the left inequality in (1.8) if and only if $f=f_{0}^{-}$
a.e. on $[-\pi,\pi]$, where $f_{0}^{-}=M-(M-m)\,\chi_{I(a_{m}^{-},l^{-})}$
with $l^{-}=\pi-l$ and $a_{m}^{-}=x_{0}(1-l^{-}c_{\alpha})$ if
$x_{0}(1-l^{-}c\alpha)<\pi-l^{-}$ and $a_{m}^{-}=\pi-l^{-}$ otherwise.

It would be also useful to know how warmer a fixed spot $x_{0}$ could be
compared to the edges of the rod. The answer to this question is the
following.

######  Theorem 3.

Let $u_{f}$ solves the Robin problem for $f\in\mathcal{F}(m,M,s)$ and
$\alpha>0$ and let $x_{0}\in[-\pi,\pi]$ be fixed. Then

$u_{f}(x_{0})-u_{f}(-\pi)\leq
m(\eta_{\alpha}(x_{0})-\eta_{\alpha}(-\pi))+(M-m)\tau_{\alpha}(x_{0},l),$
(1.9)

where $l=\pi(s-m)/(M-m)$, the function $\eta_{\alpha}(x)$ is defined by
equation (2.5) and the function $\tau_{\alpha}(x,l)$ is defined by equation
(3.5).

Equality holds in (1.9) if and only if $f=f_{e}$ a.e. on $[-\pi,\pi]$, where
$f_{e}=m+(M-m)\,\chi_{I(a_{e},l)}$ with $l$ defined above and
$a_{e}=a_{e}(x,l)$ defined in equation (3.4).

The main results of [5] stated in Theorems 1.3 and 1.5 concern the comparison
principles for heating problems with the Robin and Newmann boundary
conditions, respectively. In these problems, the extremal distribution of heat
is symmetric with respect to the center of the rod. With this symmetry, the
authors of [5] were able to use symmetrization methods due to G. Talenti [9]
and A. Baernstein II [2] to prove their theorems. We want to mention here that
S. Abramovich in her paper [1] published in 1975 already used the
symmetrization method due to G. Pólya and G. Szegö [7] to prove interesting
results on the eigenvalues of the differential system
$y^{\prime\prime}(x)+\lambda p(x)y(x)=0$, $y(\pm 1)=0$ for the function
$p(x)\geq 0$ defined on the string $(-1,1)$. More recently, the symmetrization
method similar to the one used in [5] in combination with the polarization
technique was used in [3] to study several problems on heat distribution in
the cylindrical pipes heated along various regions on the surface area.

We stress here that the extremal distributions of heat in our Theorems 1, 2
and 3 are not symmetric with respect to the center of the rod, in general.
Thus, the classical symmetrization technique cannot be applied in these
problems while certain versions of polarization technique used in [3] still
can be applied.

##  2\. Heating a fixed spot by a single interval.

Suppose that the Robin rod $[-\pi,\pi]$ is heated with unit density along the
interval $I=I(a,l)$ centered at the point $a\in(-\pi,\pi)$ with length $2l$
such that $-\pi\leq a-l<a+l\leq\pi$. Thus, we assume here that $f=\chi_{I}$.
Using the integral representation (1.4) for the Robin temperature with the
Green’s function given by (1.5), we evaluate
$u_{\chi_{I}}=u_{\chi_{I(a,l)}}(x,\alpha)$ as follows:

$\displaystyle u_{\chi_{I}}$ $\displaystyle=$
$\displaystyle\int_{I}[(-1/2)c_{\alpha})xy-(1/2)|x-y|+1/(2c_{\alpha})]\,dy$
(2.4) $\displaystyle{=}$
$\displaystyle\left\\{\begin{array}[]{lr}l[(1-ac_{\alpha})x+(c_{\alpha}^{-1}-a)],&-\pi\leq
x\leq a-l,\\\
-\frac{1}{2}x^{2}+a(1-lc_{\alpha})x+\frac{l}{c_{\alpha}}-\frac{a^{2}+l^{2}}{2},&a-l<x<a+l,\\\
l[-(1+ac_{\alpha})x+(c_{\alpha}^{-1}+a)],&a+l\leq x\leq\pi.\end{array}\right.$

In particular, if the whole rod $[-\pi,\pi]$ is heated with unit density, then
$a=0$, $l=\pi$, and $u_{\chi_{[-\pi,\pi]}}(x)=\eta_{\alpha}(x)$, where

$\eta_{\alpha}(x)=-\frac{1}{2}x^{2}+\frac{\pi}{\alpha}+\frac{\pi^{2}}{2},\quad-\pi\leq
x\leq\pi.$ (2.5)

Next, we fix $x_{0}\in[0,\pi]$, $l\in(0,\pi)$, $\alpha>0$ and treat
$u_{\chi_{I}}$ as a function $F(a)$ of the variable $a\in[-\pi+l,\pi-l]$.

We have to consider the following cases:

  1. 1)

If $0\leq x_{0}\leq-\pi+2l$, then

$F(a)=-\frac{1}{2}a^{2}+x_{0}(1-lc_{\alpha})a-\frac{1}{2}x_{0}^{2}+\frac{l}{c_{\alpha}}-\frac{l^{2}}{2},\quad-\pi+l\leq
a\leq\pi-l,$

  2. 2)

If $-\pi+2l<x_{0}<\pi-2l$, then

$F(a)=\left\\{\begin{array}[]{ll}l(1-x_{0}c_{\alpha})a+l(\frac{1}{c_{\alpha}}-x_{0}),&-\pi+l\leq
a\leq x_{0}-l,\\\
-\frac{1}{2}a^{2}+x_{0}(1-lc_{\alpha})a-\frac{1}{2}x_{0}^{2}+\frac{l}{c_{\alpha}}-\frac{l^{2}}{2},&x_{0}-l\leq
a\leq x_{0}+l,\\\
-l(1+x_{0}c_{\alpha})a+l(\frac{1}{c_{\alpha}}+x_{0}),&x_{0}+l\leq
a\leq\pi-l.\end{array}\right.$

  3. 3)

If $\max\\{-\pi+2l,\pi-2l\\}\leq x_{0}\leq\pi$, then

$F(a)=\left\\{\begin{array}[]{ll}l(1-x_{0}c_{\alpha})a+l(\frac{1}{c_{\alpha}}-x_{0}),&-\pi+l\leq
a\leq x_{0}-l,\\\
-\frac{1}{2}a^{2}+x_{0}(1-lc_{\alpha})a-\frac{1}{2}x_{0}^{2}+\frac{l}{c_{\alpha}}-\frac{l^{2}}{2},&x_{0}-l\leq
a\leq\pi-l.\end{array}\right.$

A simple argument, left to the interested reader, shows that, in all three
cases, $F(a)$ is positive and concave on the interval $[-\pi+l,\pi-l]$ and
takes its minimal value $\mu_{\alpha}=\mu_{\alpha}(x_{0},l)$ at $a=-\pi+l$.
Evaluating $\mu_{\alpha}=F(-\pi+l)$, we find

$\mu_{\alpha}=\left\\{\begin{array}[]{l}l[((\pi-l)c_{\alpha}-1)x_{0}+\frac{1}{c_{\alpha}}+l-\pi],{\mbox{\
if $-\pi+2l<x_{0}\leq\pi$,}}\\\
-\frac{1}{2}x_{0}^{2}-(\pi-l)(1-lc_{\alpha})x_{0}-l^{2}+l(\frac{1}{c_{\alpha}}+\pi)-\frac{\pi^{2}}{2},{\
\mbox{otherwise.}}\end{array}\right.$

Next, we find the maximum $\nu_{\alpha}=\nu_{\alpha}(x_{0},l)=\max F(a)$ taken
over the interval $-\pi+l\leq a\leq\pi-l$ and identify the point
$a_{m}\in[-\pi+l,\pi-l]$, where this maximum is achieved. Let $q(a)$ denote
the quadratic function as in parts 1)–3). Then $q(a)$ takes its maximum at the
point $a_{0}=x_{0}(1-lc_{\alpha})$. Notice that $0\leq a_{0}\leq x_{0}$ with
equality sign in either of these inequalities if and only if $x_{0}=0$. If
$a_{0}+l\leq\pi$, then the function $F(a)$ is defined at $a_{0}$ and takes its
maximum at this point. Thus, if $a_{0}+l\leq\pi$, then
$a_{m}=x_{0}(1-lc_{\alpha})$. If $a_{0}+l>\pi$, then $F(a)$ takes its maximum
at $a_{m}=\pi-l$. Combining these cases, we have the following equation for
the central point $a_{m}=a_{m}(x_{0},l,\alpha)$ of the heating interval of
length $2l$, which generates the maximal temperature at the point $x_{0}$:

$a_{m}=\left\\{\begin{array}[]{ll}x_{0}(1-lc_{\alpha}),&{\mbox{if
$x_{0}<\frac{\pi-l}{1-lc_{\alpha}}$,}}\\\
\pi-l,&{\mbox{otherwise.}}\end{array}\right.$ (2.6)

With these notations, we can evaluate the maximum
$\nu_{\alpha}=\nu_{\alpha}(x_{0},l)$ as follows:

$\nu_{\alpha}=\left\\{\begin{array}[]{lr}\frac{l}{c_{\alpha}}\left(1-\frac{lc_{\alpha}}{2}\right)\,\left(1-x_{0}^{2}c_{\alpha}^{2}\right),&{\mbox{if
$x_{0}<\frac{\pi-l}{1-lc_{\alpha}}$,}}\\\ \frac{1}{2}(\pi-
x_{0})^{2}+l(1-c_{\alpha}x_{0})(2\pi-l+\frac{1}{\alpha}),&{\mbox{otherwise.}}\end{array}\right.$
(2.7)

The inequality $a_{0}+l\leq\pi$ is equivalent to the inequality

$\alpha\geq\frac{x_{0}+l-\pi}{(\pi-l)(\pi-x_{0})}.$

Let us define $\alpha_{m}\geq 0$ as

$\alpha_{m}=\left\\{\begin{array}[]{ll}\frac{x_{0}+l-\pi}{(\pi-l)(\pi-
x_{0})},&{\mbox{if $x_{0}<\pi-l$,}}\\\
0,&{\mbox{otherwise.}}\end{array}\right.$ (2.8)

Now, our arguments above show that if $x_{0}\in[0,\pi)$, $l\in(0,\pi)$ and
$\alpha>\alpha_{m}$, then the function $F(a)$ achieves its maximum
$\nu_{\alpha}(x_{0},l)$, given by the first line of (2.7), at the point
$a_{m}=x_{0}(1-lc_{\alpha})$, and if $0<\alpha\leq\alpha_{m}$, then $F(a)$
achieves its maximum $\nu_{\alpha}(x_{0},l)$, given by the second line of
(2.7), at the point $a_{m}=\pi-l$.

Combining our results and using the notation introduced above, we obtain the
following lemma.

######  Lemma 1.

1) Let $x_{0}\in[0,\pi]$, $l\in(0,\pi)$, and $\alpha>0$ be fixed and let $a$
varies from $-\pi+l$ to $\pi-l$.

If $\alpha>\alpha_{m}$, then the function
$F(a)=u_{\chi_{I(a,l)}}(x_{0},\alpha)$ increases from its minimal value
$\mu_{\alpha}(x_{0},l)$ to its maximal value $\nu_{\alpha}(x_{0},l)$ as $a$
varies from $-\pi+l$ to $a_{m}=x_{0}(1-lc_{\alpha})<\pi-l$, and $F(a)$
decreases as $a$ varies from $a_{m}$ to $\pi-l$.

If $0<\alpha\leq\alpha_{m}$, then the function
$F(a)=u_{\chi_{I(a,l)}}(x_{0},\alpha)$ increases from $\mu_{\alpha}(x_{0},l)$
to $\nu_{\alpha}(x_{0},l)$ as $a$ varies from $-\pi+l$ to $a_{m}=\pi-l$.

2) Furthermore, the point $a_{m}$, where $F(a)$ takes its maximum, stays at
$\pi-l$ for $0<\alpha\leq\alpha_{m}$ and $a_{m}$ decreases from $\pi-l$ to
$\frac{\pi-l}{\pi}x_{0}$, when $\alpha$ runs from $\alpha_{m}$ to $\infty$.

3) Moreover, if $x_{0}\in[0,\pi]$ and $\alpha>0$ are fixed and $a_{m}$ is
considered as a function $a_{m}(l)$ of $l$, then if $0<l_{1}<l_{2}<\pi$, then

$x_{0}\in[a_{m}(l_{1})-l_{1},a_{m}(l_{1})+l_{1}]\subset[a_{m}(l_{2})-l_{2},a_{m}(l_{2})+l_{2}].$

##  3\. Temperature gap between a fixed spot and the edges of a rod for a
single interval.

As in the previous section, we assume that the Robin rod $[-\pi,\pi]$ is
heated with unit density along the interval $I=I(a,l)$. Let us fix
$x_{0}\in[-\pi,\pi]$, $l\in(0,\pi)$, $\alpha>0$, and consider the temperature
gap between the point $x_{0}$ and the left edge of the rod as a function of
$a\in[-\pi+l,\pi-l]$; i.e. we consider the function

$E(a)=u_{\chi_{I}}(x_{0})-u_{\chi_{I}}(-\pi).$ (3.1)

To find $E(a)$, we use equation (2.4). Depending on the values of $l$ and
$x_{0}$, we have to consider the following cases:

  1. 1)

If $-\pi\leq x_{0}\leq\min\\{-\pi+2l,\pi-2l\\}$, then

$E(a)=\left\\{\begin{array}[]{ll}-\frac{1}{2}a^{2}+(x_{0}(1-lc_{\alpha})+l(1-\pi
c_{\alpha}))a-\\\ -\frac{1}{2}x_{0}^{2}+\pi l-\frac{l^{2}}{2},&{\mbox{if
$-\pi+l\leq a\leq x_{0}+l$,}}\\\
-lc_{\alpha}(\pi+x_{0})a+l(\pi+x_{0}),&{\mbox{if $x_{0}+l\leq
a\leq\pi-l$.}}\end{array}\right.$

  2. 2)

If $-\pi+2l<x_{0}<\pi-2l$, then

$E(a)=\left\\{\begin{array}[]{ll}l(2-(\pi+x_{0})c_{\alpha})a+l(\pi-
x_{0}),&{\mbox{if $-\pi+l\leq a\leq x_{0}-l$,}}\\\
-\frac{1}{2}a^{2}+(x_{0}(1-lc_{\alpha})+l(1-\pi c_{\alpha}))a-\\\
-\frac{1}{2}x_{0}^{2}+\pi l-\frac{l^{2}}{2},&{\mbox{if $x_{0}-l\leq a\leq
x_{0}+l$,}}\\\ -lc_{\alpha}(\pi+x_{0})a+l(\pi+x_{0}),&{\mbox{if $x_{0}+l\leq
a\leq\pi-l$.}}\end{array}\right.$

  3. 3)

If $\max\\{-\pi+2l,\pi-2l\\}\leq x_{0}\leq\pi$, then

$E(a)=\left\\{\begin{array}[]{ll}l(2-(\pi+x_{0})c_{\alpha})a+l(\pi-
x_{0}),&{\mbox{if $-\pi+l\leq a\leq x_{0}-l$,}}\\\
-\frac{1}{2}a^{2}+(x_{0}(1-lc_{\alpha})+l(1-\pi c_{\alpha}))a-\\\
-\frac{1}{2}x_{0}^{2}+\pi l-\frac{l^{2}}{2},&{\mbox{if $x_{0}-l\leq
a\leq\pi-l$.}}\end{array}\right.$

  4. 4)

If $\pi-2l\leq x_{0}\leq-\pi+2l$, then for $-\pi+l\leq a\leq\pi-l$,

$E(a)=-\frac{1}{2}a^{2}+(x_{0}(1-lc_{\alpha})+l(1-\pi
c_{\alpha}))a-\frac{1}{2}x_{0}^{2}+\pi l-\frac{l^{2}}{2}.$

Next, we show that in each of the four cases above there is a unique point
$a_{e}\in[-\pi+l,\pi-l]$, where the function $E(a)$ achieves its maximum, we
call it $\tau_{\alpha}=\tau_{\alpha}(x_{0},l)$.

On can easily check that in all cases, $E^{\prime}(-\pi+l)>0$ and, in the
cases 1) and 2), $E^{\prime}(\pi-l)<0$. Since $E(a)$ is a smooth at most
quadratic function, we conclude from this that, in the cases 1) and 2), $E(a)$
achieves its maximum $\tau_{\alpha}$ at the point
$a_{e}=x_{0}+l-lc_{\alpha}(\pi+x_{0})$, $-\pi+l<a_{e}<x_{0}+l$.

In the cases 3) and 4), we find that

$E^{\prime}(\pi-l)=-\pi+2l+x_{0}-l(\pi+x_{0})c_{\alpha}.$ (3.2)

Since, in the cases 3) and 4), $-\pi+2l+x_{0}>0$, we conclude from (3.4) and
(1.5) that $E^{\prime}(\pi-l)<0$ if $\alpha>\alpha_{e}$ and
$E^{\prime}(\pi-l)\geq 0$ if $0<\alpha\leq\alpha_{e}$, where $\alpha_{e}$ is
defined as

$\alpha_{e}=\left\\{\begin{array}[]{ll}\frac{-\pi+2l+x_{0}}{(\pi-l)(\pi-
x_{0})}&{\mbox{if $x_{0}>\pi-2l$,}}\\\ 0&{\mbox{if
$x_{0}\leq\pi-2l$.}}\end{array}\right.$ (3.3)

In the first of these cases, $E(a)$ takes its maximum $\tau_{\alpha}$ at the
point $a_{e}=x_{0}+l-lc_{\alpha}(\pi+x_{0})$. In the second case, $E(a)$
achieves its maximum at the point $a_{e}=\pi-l$.

Combining these cases, we conclude that $E(a)$ achieves its maximum at the
point

$a_{e}=\left\\{\begin{array}[]{lr}\pi-l,&{\mbox{if $x_{0}>\pi-2l$ and
$0<\alpha\leq\alpha_{e}$,}}\\\
x_{0}+l-lc_{\alpha}(\pi+x_{0}),&{\mbox{otherwise.}}\end{array}\right.$ (3.4)

Calculating the values of $\tau_{\alpha}$ in the cases mentioned above, we
obtain:

$\tau_{\alpha}=\left\\{\begin{array}[]{l}-\frac{1}{2}x_{0}^{2}+\frac{\pi^{2}}{2}+(x_{0}(1-lc_{\alpha})+l(1-\pi
c_{\alpha}))(\pi-l),\\\ {\mbox{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ if $x_{0}>\pi-2l$ and $0<\alpha\leq\alpha_{0}$,}}\\\
\frac{1}{2}(x_{0}+l-lc_{\alpha}(\pi+x_{0}))^{2}-\frac{1}{2}x_{0}^{2}+\pi
l-\frac{1}{2}l^{2},{\ \ \ \ \ \mbox{otherwise.}\par}\end{array}\right.$ (3.5)

Combining our results, we obtain the following lemma.

######  Lemma 2.

Let $x_{0}\in[-\pi,\pi]$, $l\in(0,\pi)$, and $\alpha>0$ be fixed and let $a$
vary from $-\pi+l$ to $\pi-l$.

If $x_{0}$ and $l$ satisfy the inequalities in parts 1) or 2) given above in
this section or if $x_{0}$ and $l$ satisfy inequalities given in parts 3) or
4) and also $\alpha>\alpha_{e}$, where $\alpha_{e}$ is defined in (3.3), then
the function $E(a)=u_{\chi_{I}}(x_{0})-u_{\chi_{I}}(-\pi)$ increases from
$E(-\pi+l)$ to its maximal value $\tau_{\alpha}=\tau_{\alpha}(x_{0},l)$ given
by the second line in the equation (3.5) as $a$ varies from $-\pi+l$ to
$a_{e}$ defined by equation (3.4), and $E(a)$ decreases as $a$ varies from
$a_{e}$ to $\pi-l$.

If $x_{0}$ and $l$ satisfy the inequalities given in parts 3) or 4) and also
$0<\alpha\leq\alpha_{e}$, then the function $E(a)$ increases from $E(-\pi+l)$
to its maximal value $\tau_{\alpha}$ given by the first line in the equation
(3.5) as $a$ varies from $-\pi+l$ to $\pi-l$.

##  4\. Maximal temperature gap for a single interval.

The goal here is to evaluate the temperature gap ${\rm{osc}}(u_{J})$ for the
heat source $J=\delta+\chi_{I(a,l)}$, $l\in[0,\pi)$, $a\in[0,\pi-l]$,
$\delta\geq 0$. It follows from our calculations in Section 2 that

$u_{J}(x)=\delta\eta_{\alpha}(x)+u_{\chi_{I(a,l)}}(x),$ (4.1)

where $\eta_{\alpha}(x)$ is defined by (2.5) and $u_{\chi_{I(a,l)}}$ is
defined by (2.4).

An elementary calculation shows that, under our assumptions, $u_{J}(-\pi)\leq
u_{J}(\pi)$ with equality sign if and only if $a=0$. Since $u_{J}$ is concave
on $[-\pi,\pi]$ it follows that

$\min_{[-\pi,\pi]}u_{J}=u_{J}(-\pi)=\alpha^{-1}(\pi\delta+l(1-ac_{\alpha})).$
(4.2)

To find the maximum of $u_{J}$, we note that $u_{J}$ is a concave piece-wise
at most quadratic function, which achieves its maximum at the point

$x_{0}=\frac{a(1-lc_{\alpha})}{1+\delta}.$ (4.3)

Thus, $0\leq x_{0}\leq a$ with equality sign if and only if $a=0$. Evaluating
$u_{J}(x_{0})$ and simplifying, we find:

$\max_{[-\pi,\pi]}u_{J}=u_{J}(x_{0})=\frac{1}{2}\left(\frac{(1-lc_{\alpha})^{2}}{1+\delta}-1\right)\,a^{2}+\frac{\pi\delta}{\alpha}+\frac{\pi^{2}\delta}{2}+\frac{l}{c_{\alpha}}-\frac{l^{2}}{2}.$
(4.4)

Combining (4.2) and (4.4) and simplifying, we find that
${\rm{osc}}(u_{J})=H_{\alpha}$, where $H_{\alpha}=H_{\alpha}(a,l,\delta)$ is
given by

$H_{\alpha}=\frac{1}{2}\left(\frac{(1-lc_{\alpha})^{2}}{1+\delta}-1\right)\,a^{2}+\frac{l}{1+\alpha\pi}a+\pi
l+\frac{\pi^{2}\delta}{2}-\frac{l^{2}}{2}.$ (4.5)

Next we fix $l\in(0,\pi)$, $\alpha>0$, $\delta\geq 0$ and treat $H_{\alpha}$
as a function $H_{\alpha}(a)$ of $a\in[0,\pi-l]$. Since
$(1-lc_{\alpha})^{2}/(1+\delta)<1$, it follows from (4.5) that $H_{\alpha}(a)$
is concave and takes its maximum at the point

$a_{0}=\frac{(1+\delta)l}{(1+\alpha\pi)(\delta+2lc_{\alpha}-l^{2}c_{\alpha}^{2})}.$
(4.6)

Now, the question is whether or not the point $a_{0}$ given in (4.6) belongs
to the interval $[0,\pi-l]$. Notice that the function
$\delta+2lc_{\alpha}-l^{2}c_{\alpha}^{2}$ in the denominator of (4.6)
increases when $\alpha$ varies from $0$ to $\infty$. This implies that, if
$\delta$ and $l$ are fixed, then $a_{0}$, defined by (4.6) and considered as a
function $a_{0}=a_{0}(\alpha)$ of $\alpha$, decreases from
$(1+\delta)l/\delta$ to $0$ if $\delta>0$ or from $\infty$ to $0$ if
$\delta=0$, when $\alpha$ varies from $0$ to $\infty$. Therefore if
$(1+\delta)l\leq\delta(\pi-l)$, then $a_{0}\in[0,\pi-l]$ for all $\alpha>0$
and if $(1+\delta)>\delta(\pi-l)$, then there is a unique $\alpha_{0}>0$ such
that $a_{0}\in[0,\pi-l]$ for $\alpha\geq\alpha_{0}$ and $a_{0}>\pi-l$ for
$0<\alpha<\alpha_{0}$. The value of $\alpha_{0}$ is the positive root of the
quadratic equation

$(\pi^{2}\delta+2\pi l-l^{2})\alpha^{2}+\frac{2(\pi\delta+l)(\pi-l)-\pi
l(1+\delta)}{\pi-l}\,\alpha+\delta-\frac{(1+\delta)l}{\pi-l}=0,$ (4.7)

which is

$\alpha_{0}=\frac{-2\pi^{2}\delta+3\pi\delta l-\pi
l+2l^{2}+l\sqrt{(1+\delta)(\pi^{2}\delta+9\pi^{2}-16\pi
l+8l^{2})}}{2(\pi^{3}\delta-\pi^{2}\delta l+2\pi^{2}l-3\pi l^{2}+l^{3})}.$
(4.8)

When $\delta=0$ and $l=\pi/2$, we obtain $\alpha_{0}=\frac{2}{\sqrt{3}\pi}$
that is the transitional value of $\alpha$ found in Proposition 3.3 in [5].

Thus, we can define the following parameters:

$\alpha_{g}=\left\\{\begin{array}[]{ll}\alpha_{0}&{\mbox{if
$(1+\delta)l>\delta(\pi-l$),}}\\\ 0&{\mbox{if
$(1+\delta)l\leq\delta(\pi-l)$}}\end{array}\right.$ (4.9)

and

$a_{g}=\left\\{\begin{array}[]{ll}\pi-l&{\mbox{if
$(1+\delta)l>\delta(\pi-l$),}}\\\ a_{0}&{\mbox{if
$(1+\delta)l\leq\delta(\pi-l)$.}}\end{array}\right.$ (4.10)

Figure 2, which illustrates possible situations, contains graphs of ${\rm
osc}(u_{J})=H_{\alpha}(a)$ considered as a function of the parameter $a$ for
some fixed values of $l$, $\delta$ and $\alpha$. These graphs show, in
particular, that when $\alpha$ varies from $0$ to $\infty$, the central point
of the segment providing the maximal oscillation for $u_{J}$ moves toward the
center of the rod from the right. For a special case when $\delta=0$,
$l=\pi/2$, this behavior had been already observed in Proposition 3.3 in [5].

Fig 2. Graphs of ${\rm osc}(u_{J})=H_{\alpha}(a)$ as a function of $a$ for
different choices of $l$, $\delta$, and $\alpha$.

Summarizing the results of this section and evaluating the value of the
oscillation ${\rm{osc}}(u_{J})=H_{\alpha}(a,l,\delta)$ at the point $a=a_{g}$
defined in (4.10), we obtain the following.

######  Lemma 3.

Let $l\in(0,\pi)$, $a\in[0,\pi-l]$, $\alpha>0$, and $\delta\geq 0$ be fixed
and let $u_{J}$ be defined as in (4.1). Then

$\max_{a\in[0,\pi]}{\rm{osc}}(u_{J})=\Theta_{\alpha}(l,\delta),$ (4.11)

where

$\Theta_{\alpha}=\frac{1}{2}\frac{(1+\delta)l^{2}}{(\pi^{2}\delta+2\pi
l-l^{2})\alpha^{2}+2(\pi\delta+l)\alpha+\delta}+\pi
l+\frac{\pi^{2}\delta}{2}-\frac{l^{2}}{2},$ (4.12)

when $(1+\delta)l<\delta(\pi-l)$ and $\alpha>0$ or
$(1+\delta)l\geq\delta(\pi-l)$ and $\alpha>\alpha_{g}$, and

$\Theta_{\alpha}=\frac{1}{2}\frac{(l^{2}c_{\alpha}^{2}-2lc_{\alpha}-\delta)(\pi-l)^{2}}{1+\delta}+\frac{l(\pi-l)}{1+\alpha\pi}+\pi
l+\frac{\pi^{2}\delta}{2}-\frac{l^{2}}{2},$ (4.13)

when $(1+\delta)l\geq\delta(\pi-l)$ and $0<\alpha\leq\alpha_{g}$.

Furthermore, if $l\in(0,\pi)$, $\alpha>\alpha_{g}$, and $\delta\geq 0$ are
fixed, then ${\rm{osc}}(u_{J})$ considered as a function of $a$ increases when
$a$ varies from $0$ to $a_{0}$ defined by (4.6) and ${\rm{osc}}(u_{J})$
decreases when $a$ varies from $a_{0}$ to $\pi-l$. If
$0<\alpha\leq\alpha_{g}$, then ${\rm{osc}}(u_{J})$ increases when $a$ varies
from $0$ to $\pi-l$.

##  5\. Approximation by step functions and continuity.

In this section, we collect auxiliary results on convergence of sequences of
solutions $u_{f_{n}}$ of the Robin problem, on approximation of $u_{f}$ by
sequences $u_{f_{n}}$ with piece-wise constant functions $f_{n}$ and on
continuity of $u_{f_{n}}$ as a function of the parameters of approximants.
These results will be used to prove our main theorems. We start with the
following convergence lemma.

######  Lemma 4.

If $f_{n}\in\mathcal{F}(m,M,s)$, $n=1,2,\dots$, and $f_{n}\to f$
($n\to\infty$) a.e. on $[-\pi,\pi]$, then $u_{f_{n}}\to u_{f}$ uniformly on
$[-\pi,\pi]$ and for some subsequence,

$\lim_{k\to\infty}{\rm osc}(u_{f_{n_{k}}})={\rm osc}(u_{f}).$ (5.1)

_Proof._ By the Dominated Convergence Theorem, $f_{n}$ converges to $f$ in
$L^{1}$. By (1.4), for every $x\in[-\pi,\pi]$,

$\displaystyle|u_{f_{n}}(x)-u_{f}(x)|$ $\displaystyle=$
$\displaystyle\left|\int_{-\pi}^{\pi}G(x,y)(f_{n}(y)-f(y))dy\right|$
$\displaystyle\leq$ $\displaystyle\max_{[-\pi,\pi]^{2}}G\;\;\|f_{n}-f\|_{1}.$

So, $u_{f_{n}}$ converges uniformly to $u_{f}$ on $[-\pi,\pi]$.

For each $n\in\mathbb{N}$, let $x_{n}$, $\tilde{x}_{n}$ be points of minimum
and maximum of $u_{f_{n}}$, respectively. Choose a subsequence such that
$x_{n_{k}}\to x_{o}$ and $\tilde{x}_{n_{k}}\to\tilde{x}_{o}$. By a standard
property of uniform convergence (see e.g. [4, Problems 3.1.18, 3.1.23]),

$u_{f_{n_{k}}}(x_{n_{k}})\to
u_{f}(x_{o}),\;\;\;\;\;\;u_{f_{n_{k}}}(\tilde{x}_{n_{k}})\to
u_{f}(\tilde{x}_{o}),\;\;\;\;\;\;(k\to\infty).$

Therefore, for every $x\in[-\pi,\pi]$,

$u_{f}(x)=\lim_{k}u_{f_{n_{k}}}(x)\geq\lim_{k}u_{f_{n_{k}}}(x_{n_{k}})=u_{f}(x_{o}).$

So $u_{f}(x_{o})=\min u_{f}$. Similarly $u_{f}(\tilde{x}_{o})=\max u_{f}$. It
follows that

${\rm
osc}(u_{f_{n_{k}}})=u_{f_{n_{k}}}(\tilde{x}_{n_{k}})-u_{f_{n_{k}}}(x_{n_{k}})\to
u_{f}(\tilde{x}_{o})-u_{f}(x_{o})={\rm osc}(u_{f}).$

The proof is complete. $\Box$

For $n,k\in\mathbb{N}$, $1\leq k\leq n$, let
$I_{n,k}=[-\pi+2\pi(k-1)/n,-\pi+2\pi k/n]$. Thus, the intervals
$I_{n,1},\ldots,I_{n,n}$ constitute a partition of the interval $[-\pi,\pi]$
into $n$ subintervals of equal length. We need the following approximation
result.

######  Lemma 5.

Let $f\in\mathcal{F}(m,M,s)$ and $\alpha>0$. Then for every
$n,k\in\mathbb{N}$, $1\leq k\leq n$, there are constants $c_{n,k}$, $m\leq
c_{n,k}\leq M$, such that $f_{n}=\sum_{k=1}^{n}c_{n,k}\chi_{I_{n,k}}$
satisfies the following:

  1. 1)

$m\leq f_{n}\leq f_{n+1}\leq M$, $f_{n}\to f$ a.e. on $[-\pi,\pi]$,
$\|f_{n}\|_{L^{1}}\to\|f\|_{L^{1}}=2\pi s$.

  2. 2)

$u_{f_{n}}(x)\to u_{f}(x)$ uniformly on $[-\pi,\pi]$.

_Proof._ Let $\varepsilon>0$. By a standard approximation result (see e.g. [8,
Theorem 3.14]), there exists a continuous function $g$ on $[-\pi,\pi]$ such
that $\|f-g\|_{L^{1}}<\varepsilon/2$. We can demand that $m\leq g\leq M$. For
$n\in\mathbb{N}$ and $k=1,2,\dots,n$, set

$c_{n,k}=\min\\{g(x):x\in
I_{n,k}\\}\;\;\;\;\;\hbox{and}\;\;\;\;\;f_{n}=\sum_{k=1}^{n}c_{n,k}\chi_{I_{n,k}}.$

Since $f_{n}\leq g$ and $g$ is Riemann integrable,

$\|f_{n}-g\|_{L^{1}}=\int_{[-\pi,\pi]}(g-f_{n})\to 0\;\;(n\to\infty).$

Hence $f_{n}\to g$ in $L^{1}$. So $\|f_{n}-g\|_{L^{1}}<\varepsilon/2$ for all
sufficiently large $n$. It follows that $f_{n}\to f$ in $L^{1}$. Moreover,
$(f_{n})$ is an increasing sequence of functions. Therefore, $f_{n}\to f$ a.e.
on $[-\pi,\pi]$. The other assertions come easily from Lemma 4. $\Box$

Let $n\in\mathbb{N}$, $m,M,s\in\mathbb{R}$, $0\leq m\leq s\leq M$, and
$\alpha>0$ be fixed. Let $K_{n}=K_{n}(m,M,s)$ denote the compact set of points

$(\overline{t},\overline{c})=(t_{1},t_{2},\ldots,t_{n},c_{1},\ldots,c_{n})\in\mathbb{R}^{2n}$

such that $-\pi=t_{0}\leq t_{1}\leq\ldots\leq t_{n}=\pi$, $0\leq c_{k}\leq
M-m$ and

$\sum_{k=1}^{n}c_{k}(t_{k}-t_{k-1})=2\pi(s-m).$

For $(\overline{t},\overline{c})\in K_{n}$ and $x\in[-\pi,\pi]$, consider the
function $f_{\overline{t},\overline{c}}\in\mathcal{F}(m,M,s)$ defined as

$f_{\overline{t},\overline{c}}(x)=m+\sum_{k=1}^{n}c_{k}\chi_{I_{k}}(x),\;\;\hbox{where}\;\;I_{k}=[t_{k-1},t_{k}].$

Then the solution $u_{f_{\overline{t},\overline{c}}}(x)$ of the Robin problem
is a linear combination of elementary functions as in the equation (2.4). This
immediately implies the following continuity lemma.

######  Lemma 6.

Let $u_{f_{\overline{t},\overline{c}}}(x)$ be the solution to the Robin
problem considered as a function of $x\in[-\pi,\pi]$ and
$(\overline{t},\overline{c})\in K_{n}$. Then
$u_{f_{\overline{t},\overline{c}}}(x)$ is continuous on the compact set
$[-\pi,\pi]\times K_{n}$.

In particular, if $E_{1}$ is a compact subset of $[-\pi,\pi]$, $E_{2}$ is a
compact subset of $K_{n}$, then there are points in $E_{1}\times E_{2}$, where
$u_{f_{\overline{t},\overline{c}}}(x)$ achieves its minimum and maximum on
$E_{1}\times E_{2}$.

##  6\. Main proofs.

Now we are ready to present proofs of our main results. We start with the
proof of Theorem 2.

Proof of Theorem 2. Let $f\in\mathcal{F}(m,M,s)$. First, we prove the right
inequality in (1.8). It follows from Lemma 5 that, for every $\varepsilon>0$,
there exists a piece-wise constant function
$f_{n,\overline{c}^{\prime}}=m+\sum_{k=1}^{n}c_{n,k}^{\prime}\chi_{I_{n,k}}\in\mathcal{F}(m,M,s_{1})$,
where $\overline{c}^{\prime}=(c^{\prime}_{n,1},\ldots,c^{\prime}_{n,n})$, such
that $0\leq c_{n,k}^{\prime}\leq M-m$, $|s_{1}-s|<\varepsilon$ and, for the
given $x_{0}\in[-\pi,\pi]$,

$|u_{f_{n,\overline{c}^{\prime}}}(x_{0})-u_{f}(x_{0})|<\varepsilon.$ (6.1)

We note here that the integer $n$ in the definition of the function
$f_{n,\overline{c}^{\prime}}$ can be chosen as large as we need for our proof.
Indeed, for any integer $j\geq 1$, consider the partition of $[-\pi,\pi]$ into
the intervals $I_{nj,s}$, $1\leq s\leq nj$. Then consider
$\overline{c}^{\prime\prime}=(c_{nj,1}^{\prime\prime},\ldots,c_{nj,nj}^{\prime\prime})$
such that $c_{nj,s}^{\prime\prime}=c_{n,k}^{\prime}$ if $I_{nj,s}\subset
I_{n,k}$. With these notations, we have
$f_{nj,\overline{c}^{\prime\prime}}=f_{n,\overline{c}^{\prime}}$ a.e. on
$[-\pi.\pi]$.

Next, for $n$ sufficiently large, we consider
$f_{\overline{c}}(x_{0})=m+\sum_{k=1}^{n}c_{k}\chi_{I_{n,k}}(x_{0})$ and
$u_{f_{\overline{c}}}(x_{0})$ as functions of
$\overline{c}=(c_{1},\ldots,c_{n})$, assuming that $\overline{c}$ varies over
the compact set defined by the following conditions:

$0\leq c_{k}\leq M-m,\quad\frac{1}{n}\sum_{k=1}^{n}c_{k}=s_{1}-m.$ (6.2)

It follows from the continuity Lemma 6, that there is
$\overline{c}^{*}=(c_{1}^{*},\ldots,c_{n}^{*})$, satisfying conditions (6.2),
such that

$\max u_{f_{\overline{c}}}(x_{0})\leq u_{f_{\overline{c}^{*}}}(x_{0}),$ (6.3)

where the maximum is taken over all $\overline{c}$ satisfying conditions
(6.2).

Let $a^{*}=a_{m}(\frac{\pi}{n})$ denote the center of the interval $I$ of
length $\frac{2\pi}{n}$, which provides the maximum to the function
$u_{\chi_{I}}(x)$ at the point $x=x_{0}$. We recall that
$a^{*}=a_{m}(\frac{\pi}{n})$, given by equation (2.6), is such that $0\leq
a^{*}\leq x_{0}\leq a^{*}+\frac{\pi}{n}$.

Suppose that the centers of the intervals $I_{n,k}$ and $I_{n,k+1}$ lie in the
interval $[-\pi,a^{*}]$. Under these conditions, we claim that either
$c_{k}^{*}=0$ or $c_{k+1}^{*}=M-m$. Indeed, if $c_{k}^{*}>0$ and
$c_{k+1}^{*}<M-m$ then, for all sufficiently small $\delta>0$,
$\widetilde{f}=f_{\overline{c}^{*}}-\delta\chi_{I_{n,k}}+\delta\chi_{I_{n,k+1}}\in\mathcal{F}(m,M,s_{1})$.
Furthermore, it follows from Lemma 1 that
$u_{\chi_{I_{n,k}}}(x_{0})<u_{\chi_{I_{n,k+1}}}(x_{0})$. The latter together
with the additivity property (1.6) imply that
$u_{\widetilde{f}}(x_{0})>u_{f_{\overline{c}^{*}}}(x_{0})$, contradicting the
inequality (6.3). A similar argument shows that if the centers of the
intervals $I_{n,k}$ and $I_{n,k+1}$ lie in the interval $[a^{*},\pi]$ then
either $c_{k}^{*}=M-m$ or $c_{k+1}^{*}=0$.

Using these observations, we conclude that, if the integer $n$ is large
enough, then there are integers $k_{1},k_{2}$, $1\leq k_{1}<k_{2}\leq n$, such
that $c_{j}^{*}=0$ for $1\leq j<k_{1}$ and $k_{2}<j\leq n$, $c_{j}^{*}=M-m$
for $k_{1}<j<k_{2}$, and $0\leq c_{j}^{*}<M-m$, for $j=k_{1}$ and $j=k_{2}$.

Since $f_{n,\overline{c}^{\prime}}\in\mathcal{F}(m,M,s_{1})$, it follows from
(6.3) and the additivity property (1.6) that

$u_{f_{n,\overline{c}^{\prime}}}(x_{0})\leq
u_{f_{\overline{c}^{*}}}(x_{0})<u_{\widehat{f}}(x_{0}),$ (6.4)

where $\widehat{f}=m+(M-m)\chi_{\widehat{I}}$ and $\widehat{I}=[-\pi+2\pi
k_{1}/n,-\pi+2\pi k_{2}/n]$.

Let $l_{1}=\pi(s_{1}-m)/(M-m)$ and let $\widehat{l}$ denote the half length of
the interval $\widehat{I}$. Since $f^{*}\in F(m,M,s_{1})$, it follows from our
construction of $\widehat{I}$ that

$l_{1}\leq\widehat{l}\leq l_{1}+\frac{2\pi}{n}.$ (6.5)

It follows from Lemma 1 and equation (6.5) that

$u_{\chi_{\widehat{I}}}(x_{0})\leq\nu_{\alpha}(x_{0},\widehat{l})\leq\nu_{\alpha}(x_{0},l_{1})+O(1/n),$
(6.6)

where $\nu_{\alpha}(x,l)$ is defined by (2.7). The latter inequality together
with equation (2.5) and the additivity property (1.6) implies the following:

$u_{\widehat{f}}(x_{0})\leq\eta_{\alpha}(x_{0})m+(M-m)\nu_{\alpha}(x_{0},l_{1})+O(1/n).$
(6.7)

Combining (6.1), (6.4) and (6.7), we conclude that

$u_{f}(x_{0})\leq
u_{f_{n,\overline{c}^{\prime}}}(x_{0})+\varepsilon\leq\eta_{\alpha}(x_{0})m+(M-m)\nu_{\alpha}(x_{0},l)+\delta(\varepsilon,n),$
(6.8)

where $\delta(\varepsilon,n)\to 0$ when $\varepsilon\to 0$ and $n\to\infty$.
Since $\varepsilon>0$ in equation (6.1) can be chosen arbitrarily small and
the integer $n$ can be chosen arbitrarily large, (6.8) implies the second
inequality in (1.8).

To prove uniqueness, we assume that $f$ does not coincide with
$f_{0}^{+}=m+(M-m)\chi_{I(a_{m},l)}$ on a set of positive measure. Since $f\in
L^{1}$, there are “density points” $x_{1}$ and $x_{2}$, such that
$f_{0}^{+}(x_{1})=M$ and $f(x)<M-\varepsilon$ on a subset $E_{1}$ of some
small interval $I_{1}$ centered at $x_{1}$, $f_{0}^{+}(x_{2})=m$ and
$f(x)>m+\varepsilon$ on a subset $E_{2}$ of some small interval $I_{2}$
centered at $x_{2}$. Let $E=E_{1}\cap(E_{2}+(x_{1}-x_{2}))$. Since $x_{1}$ and
$x_{2}$ are density points for $f\in L^{1}$, the one-dimensional Lebesgue
measure of $E$ is strictly positive. We may assume without loss of generality
that $E$ and $E-(x_{1}-x_{2})$ either both lie on the interval $[-\pi,a_{m}]$
or both lie on the interval $[a_{m},\pi]$. Under these assumptions, it follows
from the monotonicity properties of Lemma 1 that
$u_{\chi_{E}}(x_{0})>u_{\chi_{E+(x_{1}-x_{2})}}(x_{0})$. Therefore, replacing
$f$ by $\widetilde{f}=f-\varepsilon\chi_{E-(x_{1}-x_{2})}+\varepsilon\chi_{E}$
with $\varepsilon>0$ small enough, we obtain a function
$\widetilde{f}\in\mathcal{F}(m,M,s)$ such that
$u_{f}(x_{0})<u_{\widetilde{f}}(x_{0})$. Since
$\widetilde{f}\in\mathcal{F}(m,M,s)$, the right inequality in (1.8) holds true
for $u_{\widetilde{f}}$. Therefore, for the function $u_{f}$ the right
inequality in (1.8) holds with the sign of strict inequality.

To prove the left inequality in (1.8), we assume that $f\in\mathcal{F}(m,M,s)$
and consider the function $f^{-}=M+m-f$. We have $m\leq f^{-}\leq M$ and
$\|f^{-}\|_{L^{1}}=2\pi s^{-}$ with $s^{-}=M+m-s$, $m<s^{-}<M$. Thus,
$f^{-}\in\mathcal{F}(m,M,s^{-})$ and therefore, by our proof above,

$u_{f^{-}}(x_{0})\leq\eta_{\alpha}(x_{0})m+(M-m)\nu_{\alpha}(x_{0},l^{-}),$
(6.9)

where $l^{-}=\pi(s^{-}-m)/(M-m)=\pi-l$. Since $f+f^{-}=M+m$, we have

$u_{f}(x)+u_{f^{-}}(x)=\int_{-\pi}^{\pi}G(x,y)(f(y)+f^{-}(y))\,dy=(M+m)\eta_{\alpha}(x).$
(6.10)

Combining equations (6.9) and (6.10), we obtain the left inequality in (1.8).

Furthermore, (1.8) holds with the equality sign in the left inequality if and
only if (6.9) holds with the sign of equality. We proved above that the latter
holds if and only if $f^{-}=m+(M-m)\chi_{I(a_{m}^{-},l^{-})}$ a.e. on
$[-\pi,\pi]$, which implies that the sign of equality in the left inequality
in (1.9) occurs if and only if $f=f_{0}^{-}$ a.e. on $[-\pi,\pi]$. $\Box$

The structure of proofs of Theorems 1 and 3 presented below is the same as in
the proof of Theorem 2. Basically, to prove these theorems, we use the same
arguments as in the proof of Theorem 2 with minor changes. Below, we sketch
these proofs emphasizing these minor changes.

Proof of Theorem 3. First, given $\varepsilon>0$, we approximate
$f\in\mathcal{F}(m,M,s)$ with
$f_{n,\overline{c}^{\prime}}\in\mathcal{F}(m,M,s_{1})$ such that
$|s_{1}-s|<\varepsilon$ and

$|(u_{f_{n,\overline{c}^{\prime}}}(x_{0})-u_{f_{n,\overline{c}^{\prime}}}(-\pi))-(u_{f}(x_{0})-u_{f}(-\pi))|<\varepsilon.$
(6.11)

Then, using the continuity Lemma 6, we find $f^{*}\in\mathcal{F}(m,M,s_{1})$
such that

$\max\\{u_{f_{\overline{c}}}(x_{0})-u_{f_{\overline{c}}}(-\pi)\\}\leq
u_{f_{\overline{c}^{*}}}(x_{0})-u_{f_{\overline{c}^{*}}}(-\pi),$ (6.12)

where the maximum is taken over all $\overline{c}$ satisfying conditions
(6.2).

Next, performing tricks with the intervals $I_{n,k}$ and $I_{n,k+1}$ as we did
in the proof of Theorem 2 and using Lemma 2, we obtain a function
$\widehat{f}=m+(M-m)\chi_{\widehat{I}}$, where $\widehat{I}=[-\pi+2\pi
k_{1}/n,-\pi+2\pi k_{2}/n]$ with appropriate $k_{1}$ and $k_{2}$, such that
the following holds:

$\displaystyle u_{f_{\overline{c}^{*}}}(x_{0})$ $\displaystyle-$
$\displaystyle
u_{f_{\overline{c}^{*}}}(-\pi)<u_{\widehat{f}}(x_{0})-u_{\widehat{f}}(-\pi)\leq$
$\displaystyle
m(\eta_{\alpha}(x_{0})-\eta_{\alpha}(-\pi))+(M-m)\tau_{\alpha}(x_{0},l_{1})+O(1/n).$

Since $\varepsilon>0$ in (6.11) can be taken arbitrarily small and the integer
$n$ in (6) can be taken arbitrarily large, combining (6.11), (6.12) and (6),
we obtain the inequality in (1.9).

The proof of the uniqueness statement of Theorem 3, is almost identical with
the uniqueness proof of Theorem 2. Namely, given $f\in\mathcal{F}(m,M,s)$, we
use our “two density points argument”, to construct a function
$\widetilde{f}=f-\varepsilon\chi_{E-(x_{1}-x_{2})}+\varepsilon\chi_{E}\in\mathcal{F}(m,M,s)$,
with $\varepsilon>0$ small enough, such that
$u_{f}(x_{0})-u_{f}(-\pi)<u_{\widetilde{f}}(x_{0})-u_{\widetilde{f}}(-\pi)$.
Since $\widetilde{f}\in\mathcal{F}(m,M,s)$, the inequality in (1.9) holds true
for $u_{\widetilde{f}}$. Therefore, for the function $u_{f}$, (1.9) holds with
the sign of strict inequality. $\Box$

Proof of Theorem 1. It follows from Lemma 5 that, for given $f\in F(m,M,s)$
and $\varepsilon>0$ arbitrarily small, there exists a piece-wise constant
function
$f_{n,\overline{c}^{\prime}}=m+\sum_{k=1}^{n}c_{n,k}^{\prime}\chi_{I_{n,k}}\in\mathcal{F}(m,M,s_{1})$,
where $\overline{c}^{\prime}=(c^{\prime}_{n,1},\ldots,c^{\prime}_{n,n})$, such
that $0\leq c_{n,k}^{\prime}\leq M-m$, $|s_{1}-s|<\varepsilon$ and

$|{\rm osc}(u_{f_{n,\overline{c}^{\prime}}})-{\rm osc}(u_{f})|<\varepsilon.$
(6.14)

Here, we assume once more, that the integer $n$ is chosen as large as we need
for our proof.

Then, using the continuity Lemma 6, we find
$f_{\overline{c}^{*}}=m+\sum_{k=1}^{n}c_{n,k}^{*}\chi_{I_{n,k}}\in\mathcal{F}(m,M,s_{1})$
such that

$\max\\{{\rm osc}(u_{f_{\overline{c}^{\prime}}})\\}\leq{\rm
osc}(u_{f_{\overline{c}^{*}}}),$ (6.15)

where the maximum is taken over all $\overline{c}$ satisfying conditions
(6.2). Since $u_{f_{\overline{c}^{*}}}$ is concave on $[-\pi,\pi]$, we may
assume without loss of generality that ${\rm
osc}(u_{f_{\overline{c}^{*}}})=u_{f_{\overline{c}^{*}}}(x_{0})-u_{f_{\overline{c}^{*}}}(-\pi)$
for some $x_{0}\in(-\pi,\pi]$. If this is not the case, we replace
$f_{\overline{c}^{*}}(x)$ with $f_{\overline{c}^{*}}(-x)$.

Next, we consider the temperature gap
$E(a)=u_{\chi_{I(a,l_{n})}}(x_{0})-u_{\chi_{I(a,l_{n})}}(-\pi)$ for the
interval $I(a,l_{n})$ centered at $a$ with half length $l_{n}=\pi/n$. As we
have shown in Section 3, there is a unique point $a^{*}=a_{e}$ with $a_{e}$
given by equation (3.4), where $E(a)$ achieves its maximum $\tau_{\alpha}$
given by equation (3.5).

If the centers of the intervals $I_{n,k}$ and $I_{n,k+1}$ both lie in the
interval $[-\pi,a^{*}]$ or both lie in the interval $[a^{*},\pi]$ then arguing
as in the proof of Theorem 2 and using the monotonicity properties of Lemma 2,
we conclude that if $f_{\overline{c}^{*}}$ is maximal in the sense of equation
(6.15), then either $c_{k}^{*}=0$ or $c_{k+1}^{*}=M-m$.

The latter implies that there exists an interval $\widehat{I}=[-\pi+2\pi
k_{1}/n,-\pi+2\pi k_{2}/n]$ with half length $\widehat{l}=\pi(k_{2}-k_{1})/n$
such that $l_{1}<\widehat{l}<l_{1}+\pi/n$, where $l_{1}=\pi(s_{1}-m)/(M-m)$,
and such that for $\widehat{f}=m+(M-m)\chi_{\widehat{I}}$ we have

$u_{f_{\overline{c}^{*}}}(x_{0})-u_{f_{\overline{c}^{*}}}(-\pi)\leq
u_{\widehat{f}}(x_{0})-u_{\widehat{f}}(-\pi)\leq{\rm osc}(u_{\widehat{f}}).$
(6.16)

Now, it follows from equations (4.11), (4.12), (4.13) of Lemma 3 that

${\rm
osc}(u_{\widehat{f}})\leq(M-m)\Theta_{\alpha}(\widehat{l},\delta)=(M-m)\Theta_{\alpha}(l,\delta)+\beta(\varepsilon,n),$
(6.17)

where $\delta=m/(M-m)$ and $\beta(\varepsilon,n)\to 0$ when $\varepsilon\to 0$
and $n\to\infty$.

Finally, combining equations (6.14) –(6.17), we obtain the inequality in
(1.7).

To prove the uniqueness statement of Theorem 1, we use once more the “two
density points argument” as in the proofs of Theorems 2 and 3, to construct a
function
$\widetilde{f}=f-\varepsilon\chi_{E-(x_{1}-x_{2})}+\varepsilon\chi_{E}\in\mathcal{F}(m,M,s)$,
with $\varepsilon>0$ small enough, such that ${\rm osc}(u_{f})<{\rm
osc}(u_{\widetilde{f}})$. Since $\widetilde{f}\in\mathcal{F}(m,M,s)$, the
inequality in (1.7) holds true for $u_{\widetilde{f}}$. Therefore, for the
function $u_{f}$, (1.7) holds with the sign of strict inequality.

$\Box$

##  7\. Temperature gap in higher dimensions

One can consider a variety of higher dimensional analogs of Problem 1. Here we
present two such problems on the temperature gap in pipes $P_{L}$, which are
cylindrical domains
$P_{L}=\\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}:\,x_{1}^{2}+x_{2}^{2}<1,\,|x_{3}|<L\\}$,
$L>0$. Let $S_{L}$ denote the cylindrical boundary of $P_{L}$ and let
$D_{L}\ni(0,0,L)$ and $D_{-L}\ni(0,0,-L)$ denote the boundary disks of
$P_{L}$. By $\frac{\partial}{\partial{\rm n}}$ we denote the outward normal
derivative on $\partial P_{L}$ (for the boundary points where it is defined).

######  Problem 2.

Let $E$ be a compact subset of $P_{L}$ of given volume $V$, $0<V<2\pi L$, and
let $\alpha>0$. Suppose that $u_{E}$ is a bounded solution to the Poisson
equation

$-\Delta u=\chi_{E}\;\;\;\;\hbox{in}\;\;\;P_{L},$

with mixed Neumann-Robin boundary conditions

$\frac{\partial u}{\partial{\rm n}}=0\;\;\;\;\hbox{on}\;\;\;S_{L},$
$\frac{\partial u}{\partial{\rm n}}+\alpha u=0\;\;\;\;\hbox{on}\;\;\;D_{\pm
L}.$

Find $\max{\rm osc}(u_{E})$ over all open sets $E\subset P_{L}$ of volume $V$
and identify sets $E^{*}$ providing this maximum.

######  Problem 3.

Let $\Omega$ be a compact subset of $S_{L}$ of given area $A$, $0<A<4\pi L$
and let $\alpha>0$. Suppose that $v_{\Omega}$ is a bounded solution to the
Laplace equation

$\Delta v=0\;\;\;\;\hbox{in}\;\;\;P_{L},$

with mixed Dirichlet-Robin boundary conditions

$v=\chi_{\Omega}\;\;\;\;\hbox{on}\;\;\;S_{L},$ $\frac{\partial v}{\partial{\rm
n}}+\alpha v=0\;\;\;\;\hbox{on}\;\;\;D_{\pm L}.$

Find $\max{\rm osc}(v_{\Omega})$ over all open sets $\Omega\subset S_{L}$ of
area $A$ and identify sets $\Omega^{*}$ providing this maximum.

We assume that solutions of Problems 2 and 3 exist and are regular; for the
existence and regularity of elliptic problems with Robin boundary conditions,
we refer to [6] and references therein. Problems 2 and 3 look challenging.
Similar problems to find $\max u_{E}$ or $\max v_{\Omega}$ on $P_{L}$ instead
of ${\rm osc}(u_{E})$ in Problem 2 or ${\rm osc}(v_{\Omega})$ in Problem 3 can
be more tractable. We expect that the symmetric configurations will provide
the corresponding maxima and therefore the symmetrization methods due to
Talenti and Baernstein can be applied to solve these problems.

One can also consider analogs of Problems 2 and 3 in cylindrical domains in
$\mathbb{R}^{n}$ of any dimension $n\geq 2$.

Acknowledgments. We thank Dr. G. Sakellaris and Dr. K. Yamazaki for helpful
discussions.

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