text:
\begin{document}

\thispagestyle{empty}
\begin{center}
  ~\vskip7mm {\Large\bf Sound absorption by perforated walls\\[3mm] along
    boundaries
  }\\[8mm]

  {\large Patrizia Donato\footnotemark[1],
    Agnes Lamacz\footnotemark[2], and Ben Schweizer\footnotemark[3]}\\[5mm]

\end{center}

\footnotetext[1]{Laboratoire de Math\'ematiques, Avenue de
  l'Universit\'e BP.12, F-76801 Saint-\'Etienne-du-Rouvray, \tt
  patrizia.donato@univ-rouen.fr}

\footnotetext[2]{Fakult\"at f\"ur Mathematik, U Duisburg-Essen,
  Thea-Leymann-Stra\ss e 9, D-45127 Essen, \tt agnes.lamacz@uni-due.de}

\footnotetext[3]{Fakult\"at f\"ur Mathematik, TU Dortmund,
  Vogelspothsweg 87, D-44227 Dortmund, \tt ben.schweizer@tu-dortmund.de}

\pagestyle{myheadings} \markboth{Sound absorption by perforated walls
  along boundaries}{A. Lamacz and B. Schweizer}

\begin{center}
   \begin{minipage}[c]{0.85\textwidth}
     {\bf Abstract:} We analyze the Helmholtz equation in a complex
     domain. A sound absorbing structure at a part of the boundary
     is modelled by a periodic geometry with periodicity $\eps>0$.  A
     resonator volume of thickness $\eps$ is connected with thin
     channels (opening $\eps^3$) with the main part of the macroscopic
     domain. For this problem with three different scales we analyze
     solutions in the limit $\eps\to 0$ and find that the effective
     system can
     describe sound absorption.\\[-1mm]

    {\bf MSC:} 35B27, 78M40\\[-1mm]

    {\bf Keywords:} Helmholtz equation, sound absorbers,
    homogenization, complex domain
   \end{minipage}\\[2mm]
\end{center}

\section{Introduction}

We are interested in the mathematical analysis of a sound absorbing
structure, e.g., along the wall of a room. The sound absorber consists
of a combination of small-scale structures. For the simplest setting
one should think of a wooden plate that is attached to the wall. The
plate is attached in such a way that a thin gap remains between plate
and wall. To create the sound absorption effect, little holes are
drilled in the wood to connect the room with the thin volume behind
the plate.

In order to analyze the effects of such a structure, we define a
geometry with different small scales: The wood is modelled by a layer
of thickness $\eps>0$, the (resonator) volume behind the wood has also
a thickness of order $\eps$, the holes are distributed periodically
with periodicity $\eps$. The width of the holes is assumed to be of
order $\eps^3$; this is the scaling in which a nontrivial limit
behavior is observed. We study the Helmholtz equation in the domain
that is filled with air, using homogeneous Neumann conditions along
all boundaries. Denoting solutions by $u^\eps$, we are interested in
the behavior of $u^\eps$ in the limit $\eps\to 0$. We find two
effective systems; they describe sound waves in the volume with the
sound absorbing structure.

We derive {\em two} limit systems, since the lowest order
approximation is trivial. At order $\eps^0$, the limit problem
coincides with the original Helmholtz problem: The small structures
along the boundary have no effect. In this sense, the complex geometry
can only lead to an effect of order $\eps$. We derive the effective
system for this $O(\eps)$ deviation in Theorem \ref {thm:mainTheorem}
below. Due to $L^2$-unboundedness of relevant functions, the proof is
performed with $L^1$-based function spaces and limit measures for
pressure and flux quantities.

The interesting question from the modelling perspective is: Why can
the $O(\eps)$ deviation be relevant for sound absorption? We see the
answer in the effective equation of Theorem \ref {thm:mainTheorem}:
The effective system contains the quantity $(\alpha/(LV)) - \omega^2$,
where $\alpha, L, V$ are geometric quantities, and $\omega$ is the
frequency. When the frequency is near to $\sqrt{\alpha/(LV)}$,
resonance occurs and the solutions of the $O(\eps)$-system can be very
large.  When they are of the same order as the inverse periodicity
(i.e.: $\eps^{-1}$), then the sound absorber can have a relevant
effect. This is discussed towards the end of this introduction.

\paragraph{\bf Geometry.}

We next describe the domain $\Omega_\eps$. It consists of a volume
$\Omega_0$ and some small scale structures that are attached to one
part of the boundary of $\Omega_0$. To keep the setting simple, we
assume that $\Omega_0$ is a rectangle in $\R^2$. With the two positive
parameters $a,b>0$ we denote by the interval $I := (0,a)$ the range of
the horizontal coordinate $x_1$. The limit domain is
$$\Omega_0 := (0,a) \times (-b,0) = I \times (-b,0)\,,$$
the upper boundary of $\Omega_0$ is the set
$\Gamma_0 := I\times \{0\}$.  By slight abuse of notation we will
identify functions on $I$ with functions on $\Gamma_0$.

Attached to $\Gamma_0$ is the resonator volume, which is connected
with many thin channels to the volume $\Omega_0$.  The channels are
distributed periodically with a spacing $\eps>0$; our analysis is
concerned with the limit $\eps\to 0$.  We denote by $L>0$ and $V>0$
the relative length of the channels and the relative thickness of the
resonator volume, respectively. The parameter $\alpha>0$ denotes a
relative width of the channels.

\begin{figure}[ht]
  \centering
  \begin{tikzpicture}[scale = 0.62]
    \draw[thick] (0, -6) to (10,-6) to (10,2.5) to (0,2.5) to (0,-6) circle;

    \node[] at (0.8, 1.75) {$S_\eps$};
    \node[] at (-0.8, 0.5) {$C_\eps$};
    \node[] at (0.8, -5) {$\Omega_0$};

    \draw[->] (-1, 0.0) -- (12.5, 0.0);
    \node[] at (12.5, -.52) {$x_1$};
    \draw[->] (0.0, -6.5) -- (0.0, 3.8);
    \node[] at (0.55, 3.9) {$x_2$};

    \draw[<->] (10.7, 0.0) -- (10.7, 1.0);
    \node[] at (11.4, 0.5) {$L \eps$};

    \draw[<->] (10.7, 2.5) -- (10.7, 1.0);
    \node[] at (11.4, 1.75) {$V \eps$};

    \foreach \x in {0,...,19}
    {
      \tikzset{shift={(0.5*\x, 0)}}
      \draw[] (0.2, 0.0) to (0.5, 0.0) to (0.5,1.0)
      to (0.2,1.0) to (0.2, 0.0) circle;
    }

    \end{tikzpicture}
    \caption{\small The geometry. The complex domain $\Omega_\eps$ is
      given as the union of a limit domain $\Omega_0$ (the domain
      below the $x_1$-axis), the set of channels $C_\eps$, and the
      strip $S_\eps$ above the channels. The length of the channels is
      $L\eps$, the width of the strip $S_\eps$ is $V\eps$. The
      channels are distributed with periodicity $\eps$, the width of
      the channels is $\alpha \eps^3$.}
  \label{fig:num1}
\end{figure}

The domain $\Omega_\eps$ is constructed as the union of three sets as
described below (see Figure 1).  For simplicity we always assume
$a/\eps\in \N$. The resonator strip and the channels are
\begin{align}
  S_\eps &:= I\times (L\eps, (L+V)\eps)\,,\\
  C_\eps &:= \bigcup_{k = 0}^{(a/\eps) - 1} (k\eps, k\eps +
           \alpha\eps^3)\times [0,L\eps]\,,
\end{align}
and the domain is defined as
\begin{equation}
  \label{eq:Omega-eps}
  \Omega_\eps := \Omega_0\cup S_\eps\cup C_\eps\,.
\end{equation}
The upper boundary of $\Omega_\eps$ is
$\Gamma_\eps := I\times \{ (L+V) \eps\}$. We emphasize the fact that
three scales are involved, since the channels have the width
$\alpha\eps^3$. The total volume of the channels is of the order
(length $\times$ width $\times$ number)
$|C_\eps| \sim \eps\, \eps^3\cdot \eps^{-1} = \eps^3$.

\paragraph{\bf Main results.}
We are interested in the limit behavior of a sequence $u^\eps$
satisfying the Helmholtz equation
\begin{equation}\label{eq:HH}
  \begin{split}
    -\Delta u^\eps - \omega^2 u^\eps &= f\qquad\text{ in } \Omega_\eps\,,\\
    \del_n u^\eps &= 0\qquad\text{ on } \del\Omega_\eps\,.
  \end{split}
\end{equation}
Throughout, we assume that the frequency $\omega>0$ and the right hand
side $f\in L^2(\R^2)$ are given. To simplify calculations, we assume
that $f$ has support in $\Omega_0$.

\smallskip We first provide the following theorem in order to stress
that the limit system for \eqref {eq:HH} is trivial.

\begin{theorem}[Trivial limit equation]\label{thm:0}
  Let $(u^\eps)_{\eps>0}$ be a sequence of solutions to \eqref {eq:HH}
  for some sequence $\eps\to 0$. We assume that
  $u^\eps\in H^1(\Omega_\eps)$ is bounded and that a weak limit
  $u\in H^1(\Omega_0)$ exists,
  \begin{equation}
    \label{eq:u-converg-ass}
    u^\eps|_{\Omega_0} \weakto u \quad\text{in } H^1(\Omega_0)\,.
  \end{equation}
  Then $u$ solves the trivial limit problem
  \begin{equation}
    \begin{split}
      -\Delta u - \omega^2 u &= f\qquad\text{ in } \Omega_0\,,\\
      \del_n u &= 0\qquad\text{ on } \del\Omega_0\,.
    \end{split}\label{eq:HH-trivial}
  \end{equation}
\end{theorem}

\begin{proof}
  Let $\varphi\in C^1(\R^2)$ be an arbitrary test function.  In the
  following calculation, we use first the volume estimate
  $|\Omega_\eps\setminus\Omega_0| = O(\eps)$, then the weak form of
  \eqref{eq:HH}, and finally decompose the integral and exploit the
  boundedness of the sequence $u^\eps$:
  \begin{align*}
    \int_{\Omega_0}f\varphi
    &\leftarrow \int_{\Omega_\eps}f\varphi
    = \int_{\Omega_\eps}\nabla u^\eps\cdot\nabla \varphi
      -\omega^2\int_{\Omega_\eps}u^\eps\varphi\\
    &=\int_{\Omega_0}\nabla u^\eps\cdot\nabla \varphi
       +\int_{\Omega_\eps\setminus\Omega_0}\nabla u^\eps\cdot\nabla \varphi
       -\omega^2\int_{\Omega_0}u^\eps\varphi
       -\omega^2\int_{\Omega_\eps\setminus\Omega_0}u^\eps\varphi\\
    &\to
      \int_{\Omega_0}\nabla u\cdot\nabla \varphi-\omega^2\int_{\Omega_0}u\,\varphi
  \end{align*}
  as $\eps\to 0$.  We thus obtained the weak form of
  \eqref{eq:HH-trivial}.
\end{proof}

The interesting effect in the behavior of solutions becomes visible in
the next order in $\eps$.  We define two new functions. The first
encodes the averages of $u^\eps$ with respect to the variable $x_2$ in
the resonator strip $S_\eps$,
\begin{equation}
  \label{eq:v-eps}
  v^\eps : I \to \R,\quad
  x_1 \mapsto \frac1{\eps V} \int_{\eps L}^{\eps(L+V)} u^\eps(x_1, x_2)\, dx_2\,,
\end{equation}
and the second denotes the corrector from the trivial limit,
\begin{equation}
  \label{eq:w-eps}
  w^\eps : \Omega_0 \to \R,\quad
  w^\eps := \frac{u^\eps - u}{\eps}\,.
\end{equation}

We work with the following assumption.

\begin{assumption}\label{ass:v-w}
  For some $v\in H^1(I)$ there holds $v^\eps\weakto v$ in
  $L^2(I)$. Moreover, the sequence $w^\eps$ is bounded in
  $W^{1,1}(\Omega_0)$ and, for some $w\in W^{1,1}(\Omega_0)$,
  $w^\eps\weakto w$ weak-$*$ in $BV(\bar\Omega_0)$.  The sequence
  $u^\eps\in H^1(\Omega_\eps)$ is bounded and the vertical derivative
  of $u^\eps$ satisfies the following boundedness in the channels: For
  some constant $C>0$, that does not depend on $\eps$, holds
  \begin{align}
  \label{eq:assdel2ueps}
  &\frac{1}{\eps^2}\int_{C_\eps}|\partial_2 u^\eps|\leq C\,.
  \end{align}
\end{assumption}

The weak-$*$ convergence of $w^\eps\to w$ in $BV(\bar\Omega_0)$ is
equivalent to: $w^\eps\to w$ in $L^1(\Omega_0)$ and
$\int_{\Omega_0}\nabla w^\eps\cdot \phi\to\int_{\Omega_0}\nabla
w\cdot\phi$ for all $\phi\in C(\bar\Omega_0;\R^2)$.

\smallskip
For the heuristics of Assumption \ref{ass:v-w} we refer to Section
\ref {ssec.heuristics} below.

\begin{remark}\label{rem:lower-reg-w}
  In what follows it would be sufficient to assume that $\nabla w$ is
  a measure, which is the natural assumption in the context of weak
  $BV$-convergence.  For the sake of simplicity of notation we stick
  to the stronger assumption $w\in W^{1,1}(\Omega_0)$.
\end{remark}

We are now in a position to formulate the
main result of this article. It determines the limit equation for the
function $w$. By definition of $w$, the solution $u^\eps$ has the
expansion $u^\eps \approx u + \eps w$.

\begin{theorem}[Equations for the corrector]
  \label{thm:mainTheorem}
  Let $u^\eps$ and $u$ be as in Theorem \ref {thm:0}. Let $v^\eps$ and
  $w^\eps$ be as in \eqref {eq:v-eps} and \eqref {eq:w-eps}. Let
  Assumption \ref {ass:v-w} hold with limits $v$ and $w$. Then the
  equation for $w$ is
  \begin{equation}
    \begin{split}
      -\Delta w - \omega^2 w &= 0\quad\qquad\qquad\qquad\text{ in } \Omega_0\,,\\
      \del_n w &= V\, (\del^2_1+\omega^2) v
      \qquad \text{ on } \Gamma_0\,,\\
       \del_n w &= 0
      \qquad\qquad\qquad\quad\text{ on } \del\Omega_0\setminus\Gamma_0\,,
    \end{split}\label{eq:HH-w}
  \end{equation}
  and the equation for $v$ is
  \begin{equation}
    \label{eq:limiteq-v}
    \left(-\del_1^2 + \left(\frac{\alpha}{LV}-\omega^2\right)\right) v
    = \frac{\alpha}{LV} u|_{\Gamma_0} \,.
  \end{equation}

  The function $v$ has the regularity $v\in W^{2,1}(I)$. System \eqref
  {eq:HH-w}--\eqref {eq:limiteq-v} has to be understood in the weak
  sense: For every $\fhi\in C^1(\bar\Omega_0)$ holds
  \begin{equation}\label{eq:main-w-weak}
    \int_{\Omega_0} \nabla w\cdot \nabla\fhi
    - \omega^2 \int_{\Omega_0}  w\, \fhi
      =  - V \int_{\Gamma_0} (\del_1 v\, \del_1 \fhi
      - \omega^2 v\, \fhi ) \,,
  \end{equation}
  and for every $\psi\in C^1(\bar\Gamma_0)$ holds
  \begin{equation}
    \label{eq:limiteq-v-weak}
    \int_{\Gamma_0} \del_1 v\, \del_1 \psi
    + \int_{\Gamma_0} \left(\frac{\alpha}{LV}-\omega^2\right) v \,\psi
    = \int_{\Gamma_0} \frac{\alpha}{LV} u \,\psi \,.
  \end{equation}
  We note that \eqref {eq:limiteq-v-weak} encodes not only \eqref
  {eq:limiteq-v}, but additionally the homogeneous Neumann boundary
  condition $\del_1 v = 0$ at $\del I$.
\end{theorem}

We have formulated the limiting system in a form that shows the
existence and uniqueness of solutions for almost all frequencies
$\omega$. The limit problem for $u$ has a unique solution if $\omega$
is not an eigenvalue of the Neumann Laplace operator on
$\Omega_0$. Given $u\in H^1(\Omega_0)$ and its trace
$u|_{\Gamma_0} \in H^{1/2}(\Gamma_0)$, equation \eqref {eq:limiteq-v}
with Neumann boundary conditions $\del_1 v = 0$ at $\del I$ can be
solved for $v\in H^2(I)$. Finally, assuming again that $\omega$ is not
an eigenvalue of the Neumann Laplace operator on $\Omega_0$, we can
solve system \eqref {eq:HH-w} for $w\in H^1(\Omega_0)$. This line of
argument yields not only existence, but also uniqueness of solutions
with $w$ of class $H^1$.

We note that we required less regularity on $w$ in Assumption
\ref{ass:v-w}. Our results imply that, if the limit has the additional
regularity $w\in H^1(\Omega_0)$, then it necessarily coincides with
the unique $H^1$-solution of the limit system (for $\omega$ not an
eigenvalue of the Neumann Laplace operator of $\Omega_0$).

The limit equation \eqref{eq:limiteq-v} can be re-written as
\begin{equation}\label{eq:equiv-boundary-eq}
  (\del^2_1+\omega^2)v = \frac{\alpha}{LV}(v-u)\quad\text{on }\Gamma_0\,.
\end{equation}
The boundary condition for $w$ along $\Gamma_0$ can therefore be
expressed as $\del_n w = \frac{\alpha}{L} (v-u)$ and equation \eqref
{eq:main-w-weak} can be written as
\begin{equation*}
  \int_{\Omega_0}\nabla w\cdot\nabla\varphi -\omega^2\int_{\Omega_0}w\varphi
  = \frac{\alpha}{L} \int_{\Gamma_0} (v-u) \varphi\,.
\end{equation*}
The derivation of \eqref {eq:HH-w} is actually not difficult, we
present the proof in Proposition \ref{prop:A}. The connections between
$u$ and $v$ are more involved, we derive two relations in Propositions
\ref{prop:geometricflowrule} and \ref{prop:B}.  Theorem
\ref{thm:mainTheorem} is proved after Proposition \ref{prop:B}.

\paragraph{\bf Interpretation of the main result.}

As stressed before, the limit solution $u$ is not affected by the
small scale structures along the boundary.

Let us study the limit equation \eqref{eq:limiteq-v}. The
function $v$ depends only on the horizontal coordinate $x_1$. Let us
consider solutions of the form $v(x_1) = v_0 \sin(k x_1)$ and
$u(x_1, 0) = u_0 \sin(k x_1)$ for some real parameters
$v_0, u_0 \in \R$.  Equation \eqref{eq:limiteq-v} then reads
\begin{equation*}
  \left( k^2  + \frac{\alpha}{LV} - \omega^2\right) v_0
  = \frac{\alpha}{LV} u_0\,.
\end{equation*}
This relation implies that, for resonant frequencies $\omega$, the
factor $v_0$ can be much larger than the factor $u_0$. For small
horizontal wave numbers $k$, this occurs when $\omega$ is close to the
Helmholtz resonator frequency $\omega_H := \sqrt{\alpha/(LV)}$.

When all the functions $w, u$, and $v$ have the dependence
$\sin(kx_1)$ on $x_1$, then the problem for $w$ is a homogeneous
Helmholtz problem with the upper boundary condition
\begin{align*}
  \del_n w &= \frac{\alpha}{L} (v-u)=\frac{\alpha}{L}(v_0-u_0)\sin(k x_1)\\
           &=\frac{\alpha}{L}\left[\frac{\alpha}{LV}
             \left(k^2+\frac{\alpha}{LV}-\omega^2\right)^{-1} - 1
             \right]u_0\sin(kx_1)\,.
\end{align*}
The factor in squared brackets can be large due to resonance (small
denominator). This results in large values of the function $w$. In the
reconstruction of $u^\eps$ we obtain $u^\eps \approx u + \eps w$, and
the correction has the order
$\eps \| w\| = O(\eps ( k^2 - \omega^2 + \frac{\alpha}{LV}
)^{-1})$. Due to the resonance, this can constitute a visible (or,
better: audable) contribution even for small periodicity length
$\eps>0$.

\paragraph{\bf Literature.}

Some of the first mathematical results in the field of homogenization
regarded the derivation of limit equations for domains that are
periodically perforated, see, e.g., \cite{Cioranescu1997}. Quickly,
the interest shifted also to geometries where the perforations are
along lower dimensional manifolds, we refer to \cite {MR1655534,
  MR658023} for two early contributions. The periodic unfolding method
was adapted to this kind of problems, see \cite {MR2401689}.  For the
problem in the context of fluid mechanics, see \cite {MR884812}.

As a ``natural scaling'' we regard the setting where the periodicity
is $\eps>0$, and the typical size of the obstacles is also $\eps$ (in
every direction). This scaling was also considered in the papers \cite
{MR3324482, MR2935369, MR3109435, DHS, Schweizer-Neumann-2018}.  The
aim of these papers is to provide a thorough analysis of the Neumann
problem, for which no effects of order $\eps^0$ are induced by the
geometry. In order to derive limit equations one has to analyze higher
order effects. Progress was possible in \cite {Schweizer-Neumann-2018}
with the consequent use of $W^{1,1}$-spaces: the expansion of the
solution has natural bounds in the corresponding norms.

We emphasize that, in the natural scaling, where periodicity, width,
and the length of the channels are all of order $\eps$, no resonances
can occur. In such a setting, one can only expect that deviations from
the trivial limit solution $u$ are of order $\eps$.

We note that another scaling is used, e.g., in
\cite{Plasmonic-waves-2013, Neuss-Radu-Jaeger, MR3808152}: Here, a
structure of finite width is analyzed. For a periodicity $\eps>0$ and
a diameter of the channels of order $\eps$, the length of the channels
does not tend to $0$ as $\eps\to 0$. This scaling allows for
resonances in the longitudinal direction of the channels.  Yet another
setting of the geometry was used, e.g., in \cite {MR2151798}: One
considers ``perforations'' in the boundary or in an interface of lower
dimension. The resulting system has the character of an oscillatory
boundary condition, we mention \cite {MR2739003} as a contribution in
this vast field.

The combination of two different small length scales in the obstacles
can create resonant structures. This is well-known for the Helmholtz
resonator and it was used for an analysis of spectral properties in
\cite {Schweizer-Helmholtz-resonator}. Using the small Helmholtz
resonator as a building block, one can create resonant bulk materials,
see \cite {Lamacz-Schweizer-manyres-2017}. In that work, the
resonators are distributed in the whole volume and not only along the
boundary.  For an overview regarding resonances and homogenization in
this spirit, we mention \cite {MR3591457}.

\section{Preliminaries and proof of \eqref {eq:main-w-weak}}

\subsection{Expected orders of different quantities}
\label{ssec.heuristics}

It might be surprising that we work with $L^1$-based spaces. The
choice of the function space is important. In fact, we claim that
working only in $L^2$-based function spaces is not adequate in the
problem at hand. We note that a similar observation was made in \cite
{Schweizer-Neumann-2018}.

Let us discuss heuristically the behavior of solutions.  We expect
that $u^\eps$ has values of order $\eps^0 = 1$ everywhere, in the
domain $\Omega_0$ and in the resonator strip $S_\eps$.

Since the channels are thin, there is is only a weak connection
between the volume $\Omega_0$ and the strip $S_\eps$. There is no
reason why the values of $u^\eps$ at both ends of the channel should
be close. We can therefore expect that also the difference $v- u$ is
of order $1$. As a result, since the length of each channel is of
order $\eps$, the derivative $\del_2 u^\eps$ should be of order
$\eps^{-1}$ in the channels.

With respect to \eqref {eq:assdel2ueps} we recall that the total
volume of the channels is of the order $|C_\eps| \sim \eps^3$. We can
therefore expect that the quantity in \eqref {eq:assdel2ueps},
$\eps^{-2} \int_{C_\eps}|\partial_2 u^\eps|$, is bounded.

We note that the boundedness of $\nabla u^\eps \in L^2(\Omega_\eps)$
implies the following property of horizontal derivatives:
\begin{equation}\label{eq:assdel1ueps}
  \frac{1}{\eps}\int_{C_\eps}|\partial_1 u^\eps|
  \le \frac{1}{\eps} \| \nabla u^\eps \|_{L^2(\Omega_\eps)} |C_\eps|^{1/2}
  \le C \eps^{3/2 - 1} \to 0\,.
\end{equation}

We include the warning that $L^2$-spaces are not adequate for this
problem. We expect
$\eps^{-2} \int_{C_\eps}|\partial_2 u^\eps|^2 \sim \eps^{-2} |C_\eps|
\eps^{-2} \sim \eps^{-1} \to \infty$.  In particular, we do not expect
that $L^2$-norms of $\nabla w^\eps$ are bounded. This is why we work
in the $L^1$-family of norms.

\subsection{Proof of \eqref {eq:main-w-weak}}

In this subsection we derive that the corrector $w$ satisfies
\eqref{eq:HH-w}. More precisely, we derive the weak form \eqref
{eq:main-w-weak}.

\begin{proposition}
  \label{prop:A}
  Let the sequence $u^\eps$ be as in Theorem
  \ref{thm:mainTheorem}. Then the limit function $w$ satisfies the
  effective equation \eqref {eq:main-w-weak}.
\end{proposition}

\begin{proof}
  Let $\fhi\in C^1(\R^2)$ by an arbitrary test function.  Decomposing
  the integral over $\Omega_\eps$ into intgrals over $\Omega_0$ and
  $S_\eps\cup C_\eps$, the equation for $u^\eps$ reads
  \begin{align*}
    \int_{\Omega_0}\nabla u^\eps\cdot\nabla\fhi-\omega^2\int_{\Omega_0}u^\eps\fhi
    + \int_{S_\eps\cup C_\eps}\nabla u^\eps\cdot\nabla\fhi
    -\omega^2\int_{S_\eps\cup C_\eps}u^\eps\fhi
    = \int_{\Omega_\eps}f\fhi=\int_{\Omega_0}f\fhi\,.
  \end{align*}
  On the other hand, the equation for $u$ provides
  \begin{align*}
    \int_{\Omega_0}\nabla u\cdot\nabla\fhi-\omega^2\int_{\Omega_0}u\fhi
   =\int_{\Omega_0}f\fhi\,.
  \end{align*}
  We subtract the two equations, divide by $\eps$, and insert the
  definition $w^\eps = (u^\eps - u)/\eps$ to obtain
  \begin{equation}
  \label{eq:weakformweps}
    \int_{\Omega_0} \nabla w^\eps\cdot \nabla\fhi
    - \omega^2 \int_{\Omega_0}  w^\eps\, \fhi
    = - \frac1{\eps} \left\{ \int_{S_\eps \cup C_\eps} \nabla u^\eps\cdot \nabla\fhi
      - \omega^2 \int_{S_\eps \cup C_\eps}  u^\eps\, \fhi \right\}\,.
  \end{equation}

  We consider test functions $\fhi$ that have the regularity
  $\fhi|_{\bar\Omega_0} \in C^1(\bar\Omega_0)$, assume that they are
  independent of $x_2$ for $x_2\geq 0$ and that they satisfy
  $\del_1\fhi = 0$ in the set
  $S_\eps\cap \{x_1 < \delta \text{ or } x_1 > a - \delta\}$ for some
  $\delta>0$.

  Assumption \ref{ass:v-w} on $w^\eps$ implies that the left hand side
  of \eqref{eq:weakformweps} converges, as $\eps\to 0$,
  \begin{equation*}
    \int_{\Omega_0} \nabla w^\eps\cdot \nabla\fhi
    - \omega^2 \int_{\Omega_0}  w^\eps\, \fhi
    \to\int_{\Omega_0} \nabla w\cdot \nabla\fhi
    - \omega^2 \int_{\Omega_0}  w\, \fhi\,.
  \end{equation*}

  We can use Assumption \ref{ass:v-w} also to calculate the right hand
  side of \eqref{eq:weakformweps}. All integrals over $C_\eps$ vanish
  in the limit $\eps\to 0$ because of boundedness of $u^\eps$ and
  $\nabla u^\eps$ in $L^2(\Omega_\eps)$. For one of the remaining two
  integrals, we use $\del_2 \fhi =0$ in $S_\eps$ and an integration by
  parts to find
  \begin{align*}
    -\frac1{\eps} \int_{S_\eps} \nabla u^\eps\, \nabla\fhi
    = -\frac1{\eps} \int_{S_\eps} \del_1 u^\eps\, \del_1\fhi
    =\frac{1}{\eps}\int_{S_\eps}u^\eps\,\del^2_1\fhi
    \to V \int_I v\, \del^2_1 \fhi
    = - V \int_I \del_1 v\, \del_1 \fhi\,.
  \end{align*}
  The last integral satisfies
  \begin{align*}
    \frac1{\eps} \int_{S_\eps}  u^\eps \fhi \to V \int_I  v  \fhi\,,
  \end{align*}
  as $\eps\to 0$.  Combining these limits, we arrive at
  \begin{align*}
    &\int_{\Omega_0} \nabla w\cdot \nabla\fhi
    - \omega^2 \int_{\Omega_0}  w\, \fhi
      =  V \int_I (-\del_1 v(x_1)\, \del_1 \fhi(x_1,0)
      + \omega^2 v(x_1) \fhi(x_1,0))\,dx_1 \,.
  \end{align*}
  By density of functions $\fhi$ as above, this relation holds for all
  $\fhi\in C^1(\bar\Omega_0)$. We have obtained \eqref
  {eq:main-w-weak}.
\end{proof}

\subsection{A flux quantity}

Relation \eqref{eq:limiteq-v} between $u$ and $v$ requires much more
involved arguments. We introduce an additional quantity, the vertical
flux $j^\eps$ and its limit $j_*$. We set
\begin{equation}
  \label{eq:currentdef}
  j^\eps(x) := \frac{1}{L\eps^2}\, \del_2 u^\eps(x)\, \one_{C_\eps}(x)\,,
\end{equation}
where $\one_{C_\eps}$ is the characteristic function of the channels,
$\one_{C_\eps}(x) = 1$ for $x\in C_\eps$ and $\one_{C_\eps}(x) = 0$
for $x\not\in C_\eps$. The quantity $j^\eps$ measures, in a rescaled
fashion, the $x_2$-derivative of $u^\eps$ in the channels.

\begin{lemma}
  \label{lem:currentintroduction}
  Let $u^\eps$ be a sequence as in Theorem \ref {thm:mainTheorem} and
  let $j^\eps$ be as in \eqref{eq:currentdef}. Then there exists a
  subsequence $\eps\to 0$ and a Radon measure
  $j_*\in\mathcal{M}(\R^2)$ with $\supp(j_*)\subset \bar\Gamma_0$ such
  that
  \begin{equation}
    \label{eq:j-eps-star}
    j^\eps\weakstar j_*
  \end{equation}
  in the sense of Radon measures.
\end{lemma}

\begin{proof}
  By Assumption \eqref{eq:assdel2ueps} on $\del_2 u^\eps$, the current
  $j^\eps$ is uniformly bounded in $L^1(\R^2)$:
  \begin{equation*}
    \|j^\eps\|_{L^1(\R^2)}=\frac{1}{L\eps^2}\int_{C_\eps}|\del_2 u^\eps|
    \leq \frac{C}{L}\,.
  \end{equation*}
  With the two-dimensional Lebesgue measure $\calL^2$, we can consider
  $j^\eps\, \calL^2$ as a bounded family of measures. The weak star
  compactness of Radon measures implies the existence of a subsequence
  and of a limit measure $j_*\in\mathcal{M}(\R^2)$ with \eqref
  {eq:j-eps-star}.  The measure $j_*$ is concentrated on $\bar\Gamma_0$
  since the measures $j^\eps\, \calL^2$ are supported in the channels
  $C_\eps$, hence in an $\eps$-neighborhood on $\Gamma_0$.
\end{proof}

We will use Lemma \ref {lem:currentintroduction} as follows: For every
function $\fhi\in C(\R^2)$ there holds, as $\eps\to 0$,
\begin{equation}
  \frac{1}{L\eps^2}\int_{C_\eps}\del_2 u^\eps(x)\fhi(x)\,dx
  \to \int_{\bar\Gamma_0}\fhi(x)\,dj_*(x)\,.
\end{equation}

As a preparation of one of the subsequent proofs, we note that the
arguments of Lemma \ref {lem:currentintroduction} can be repeated for
the absolute values of $j^\eps$: We consider $J^\eps := |
j^\eps|$. The measures $J^\eps \calL^2$ are a bounded family of Radon
measures. Along a subsequence $\eps\to 0$ we can therefore assume, for
some limit Radon measure $J_* \in \calM(\R^2)$ with support in
$\bar\Gamma_0$, that $J^\eps\weakstar J_*$ in the sense of Radon measures.

\section{Relations between $u$ and $v$}
\label{sec:geomflowrule}

In this section we obtain equation \eqref{eq:limiteq-v} for $u$ and
$v$. It is obtained from two other relations that involve $u$, $v$,
and the flux quantity $j_*$: The geometric flow rule \eqref
{eq:geometricflowrule} and the mass conservation \eqref
{eq:densityjstar1}. Upon eliminating the flux $j_*$, we obtain
\eqref{eq:limiteq-v}.

The first of these two new relations is the geometric flow rule and is
shown in Proposition \ref{prop:geometricflowrule}.  This geometric
rule can be perceived as follows: When $u^\eps$ has the typical value
$v$ at the upper end of the channel and the typical value $u$ at the
lower end of the channel, then the derivative has the typical value
$\del_2 u^\eps \sim (v-u)/(L\eps)$. For the integral of $j^\eps$ over
a single channel (with length $L\eps$ and width $\alpha \eps^3$) we
therefore expect to obtain
$L\eps\, \alpha \eps^3 / (L\eps^2)\cdot (v-u)/(L\eps) = (\alpha\eps/L)
(v-u)$. The factor $\eps$ denotes the periodicity. We therefore expect
a relation of the form $j_* = (\alpha/L) (v-u)$. The argument is made
precise in the following Proposition.

\begin{proposition}[Geometric flow rule]
  \label{prop:geometricflowrule}
  Let $j_*$ be as in Lemma \ref{lem:currentintroduction}, $v$ and $u$
  as in Assumption \ref{ass:v-w} and Theorem \ref{thm:0}.  Then there
  exists a density function $j\in L^1(I)$ such that
  \begin{equation}
    \label{eq:reljstarv}
    j_*(x)=j(x_1)\,\mathcal{H}^1|_{\Gamma_0}\,.
  \end{equation}
  The density satisfies
  \begin{equation}
    \label{eq:geometricflowrule}
    j(x_1)=\frac{\alpha}{L}(v(x_1)-u(x_1,0))\,.
  \end{equation}
\end{proposition}

\begin{proof}
  Once \eqref{eq:reljstarv}--\eqref{eq:geometricflowrule} are shown,
  the $L^1$-regularity of $j$ follows directly from the fact that
  $u(\cdot,0)$ and $v$ are of class $L^1(I)$.  We only have to prove
  \eqref{eq:reljstarv}--\eqref{eq:geometricflowrule}.

  Let $[c,d]\subset [0,a] = \bar I$ be an interval. Since $j_*$ is
  supported on $\bar\Gamma_0$, Proposition
  \ref{prop:geometricflowrule} is proved as soon as we can show that the
  limit measure $j_*$ satisfies
  \begin{equation}
    \label{eq:toshowgeomflowrule}
    \int_{[c,d]\times\{0\}}\,dj_*=\frac{\alpha}{L}\int_c^d (v(x_1)-u(x_1,0))\,dx_1\,.
  \end{equation}

  We will use the following function $\theta^\eps$ with large
  gradients:
  \begin{align}
    \label{eq:functionlargegradient}
    \theta^\eps(x_2):=
    \begin{cases}
      0\quad&\text{for } x_2\leq 0\,,\\
      x_2/(\eps L)\quad&\text{for } 0<x_2<\eps L\,,\\
      1\quad&\text{for } x_2\geq \eps L\,.
    \end{cases}
  \end{align}

  We want to use a localization function $\psi_\eps: [0,a]\to \R$. As
  a test function we then consider
  $\fhi^\eps(x_1,x_2) := \psi_\eps(x_1)\theta^\eps(x_2)$. The proof of
  \eqref {eq:toshowgeomflowrule} consists in calculating the quantity
  \begin{equation}
    \label{eq:defBeps}
    B_\eps:=\frac1\eps\int_{C_\eps}\nabla u^\eps\cdot \nabla\fhi^\eps
  \end{equation}
  in two different ways.

  As localization function $\psi_\eps$ we cannot use the
  characteristic function $\chi_{[c,d]} : [0,a] \to \{0,1\}$ of the
  interval $[c,d]$, since the jumps of this function can occur within
  a channel. We choose to consider all cells that touch the interval
  $[c,d]$: We define a set $\calK_\eps$ of indices as
  \begin{equation}
    \label{eq:defIeps}
    \calK_\eps :=
    \{k_1\in\Z\,|\, \eps k_1 \in [0,a-\eps]\text{ and }
    \left(k_1\eps,k_1\eps+\eps\right)\cap [c,d] \neq \emptyset\}\,.
  \end{equation}
  The number of elements of $\calK_\eps$ is of order
  $|\calK_\eps| = O(\eps^{-1})$.  We furthermore introduce
  $I^\eps_{c,d} := \bigcup_{k_1\in
    \calK_\eps}\left(k_1\eps, k_1\eps+\eps\right)$ and set
  \begin{equation}
    \psi_\eps(x_1) :=
    \begin{cases}
      1 \quad &\text{for }
      x_1 \in I^\eps_{c,d}\,,\\
      0  \quad &\text{else\,. }
    \end{cases}
  \end{equation}

  \smallskip \textit{First calculation of $B_\eps$.} We write $B_\eps$
  with the flux variable $j^\eps$ as
  \begin{align*}
    B_\eps
    &=\frac1\eps\int_{C_\eps}\nabla u^\eps\cdot \nabla\fhi^\eps
      =\frac{1}{\eps^2 L}\int_{C_\eps\cap\{x_1\in I^\eps_{c,d}\}} \del_2 u^\eps
      = \int_{\{x_1\in I^\eps_{c,d}\}} j^\eps \,d\calL^2\,.
  \end{align*}
  We claim that this implies, as $\eps\to 0$,
  \begin{equation}\label{eq:claim1-Beps}
    B_\eps \to \int_{[c,d]\times\{0\}}\,dj_*\,.
  \end{equation}
  Indeed, for every $\delta>0$ and for every $\eps< \delta$,
  there holds, as $\eps\to 0$,
  \begin{align*}
    &\left| B_\eps - \int_{\{x_1 \in [c,d] \}}\, dj_* \right|
      \le \left| \int_{\{x_1\in (0,a) \cap (c-\delta , d+\delta)\} } j^\eps \, d\calL^2
      - \int_{\{x_1\in (c-\delta , d+\delta)\} }\, dj_* \right|\\
    &\qquad\qquad
      +  \int_{\{x_1\in (c-\delta,c)\cup (d,d+\delta)\} } J^\eps\, d\calL^2
      + \left| \int_{\{x_1\in (c-\delta,c)\cup (d,d+\delta)\} } dj_* \right|\\
    &\quad \to
      \int_{\{x_1\in (c,c+\delta)\cup (d-\delta,d)\} } dJ_*
      + \left| \int_{\{x_1\in (c,c+\delta)\cup (d-\delta,d)\} } dj_* \right|\,.
  \end{align*}
  By outer regularity of the Radon measures $J_*$ and $j_*$, the right
  hand side is arbitrarily small for small $\delta>0$. This verifies
  \eqref {eq:claim1-Beps}.

  \smallskip \textit{Second calculation of $B_\eps$.} The second
  calculation of $B_\eps$ is based on a quite elementary integration
  by parts: The integral of the derivative is given by the difference
  of values at top and bottom of the channels.

  To perform the calculation, we need some additional notation.
  Recall that the microscopic channels are defined as
  $C_\eps := \bigcup_{k_1 = 0}^{a/\eps - 1} (k_1 \eps, k_1 \eps +
  \alpha\eps^3)\times [0,L\eps]$. We define the union of the lower and
  upper channel boundaries in the interval $(c,d)$ as
  \begin{equation*}
    \Gamma_\eps^U := \bigcup_{k_1\in \calK_\eps}
    (k_1\eps, k_1\eps + \alpha\eps^3)\times \{0\}
    \quad\text{ and }\quad
    \Gamma_\eps^V := \bigcup_{k_1\in \calK_\eps}
    (k_1\eps, k_1\eps + \alpha\eps^3)\times \{\eps L\}\,.
  \end{equation*}
  With this notation, an integration by parts provides
  \begin{align}
    \begin{split}
      \label{eq:Bepsfinal} B_\eps
      =\frac{1}{\eps^2 L}\int_{C_\eps\cap\{x_1\in I^\eps_{c,d}\}}\del_2 u^\eps
      =\frac{1}{\eps^2 L}\left(\int_{\Gamma_\eps^V}u^\eps-\int_{\Gamma_\eps^U}u^\eps\right)\,.
    \end{split}
  \end{align}
  It remains to determine the limit on the right hand side
  of \eqref{eq:Bepsfinal}. We will prove that
  \begin{align}
    \label{eq:meanvalueschannels1}
    &\lim_{\eps\to 0}\,\frac{1}{\eps^2} \int_{\Gamma_\eps^V}u^\eps
      = \alpha \int_{c}^d v(x_1)\,dx_1\,,\\
    \label{eq:meanvalueschannels2}
    &\lim_{\eps\to 0}\,\frac{1}{\eps^2}\int_{\Gamma_\eps^U}u^\eps
      = \alpha \int_{c}^d u(x_1,0)\,dx_1\,.
  \end{align}
  Once \eqref{eq:meanvalueschannels1}--\eqref {eq:meanvalueschannels2}
  is shown, the proof of the proposition is complete: together with
  \eqref {eq:claim1-Beps}, we obtain
  \begin{equation*}
    \int_{[c,d] \times\{0\}}\,dj_*
    =\lim_{\eps\to 0} B_\eps
    =\frac{\alpha}{L}\int_{c}^d(v(x_1)-u(x_1,0))\,dx_1\,.
  \end{equation*}
  Since $[c,d]$ was arbitrary, relations
  \eqref{eq:reljstarv}--\eqref{eq:geometricflowrule} are verified.

  \smallskip
  \textit{Verification of \eqref{eq:meanvalueschannels1}--\eqref
    {eq:meanvalueschannels2}.}  We consider the unit cell
  $Y:=(0,1)\times \left(-1,L+V\right)$ and the index set $\calK_\eps$
  of \eqref{eq:defIeps} and study the following averaged functions on
  $Y$:
  \begin{align*}
    U^\eps(y_1,y_2):=\frac{1}{|\calK_\eps|}
    \sum_{k_1\in \calK_\eps}u^\eps(\eps(k_1+y_1), \eps y_2)\quad
    \text{for }(y_1,y_2)\in Y\,.
  \end{align*}

  The (rescaled) channel in the periodicity cell $Y$ is
  $C^\eps_Y := (0,\alpha\eps^2)\times [0,L] \subset Y$. Its lower and
  upper boundary are the sets
  $\Gamma_Y^{\eps,U} := [0,\alpha\eps^2]\times\{0\}$ and
  $\Gamma_Y^{\eps,V}:= [0,\alpha\eps^2]\times\{L\}$.  The domain below
  the channel is $Y_U := (0,1)\times (-1,0)$, the domain above the
  channel is $Y_V := (0,1)\times (L,L+V)$.

  Our first aim is to prove that the restrictions $U^\eps|_{Y_U}$ and
  $U^\eps|_{Y_V}$ converge (weakly in $L^2$) to constant functions.

  Using Jensen's inequality and the fact that
  $|\calK_\eps|^{-1} = \eps/(d-c) + O(\eps^2)$, we find
  \begin{align*}
    \int_{Y_U}|U^\eps(y)|^2\,dy
    &\leq
      \frac{1}{|\calK_\eps|}\sum_{k_1\in \calK_\eps}
      \int_{Y_U}|u^\eps(\eps(y_1+k_1),\eps y_2)|^2\,dy\\
    &= \frac{1}{\eps^2|\calK_\eps|}\sum_{k_1\in
      \calK_\eps}\int_{\left(k_1\eps,k_1\eps + \eps \right)\times
      \left(-\eps,0\right)}|u^\eps(x)|^2\,dx\\
    &\leq \frac{1}{\eps^2}\left(\frac{\eps}{d-c}+O(\eps^2)\right)\int_{(c,d)\times
      (-\eps,0)}|u^\eps(x)|^2\,dx\\
    &\leq \left(\frac{1}{d-c}+O(\eps)\right)\left(\frac1\eps\int_{(c,d)\times
      (-\eps,0)}|u^\eps(x)|^2\,dx\right)\leq C\,,
  \end{align*}
  where in the last step we exploited the boundedness of $u^\eps$ in
  $H^1(\Omega_0)$: The second bracket in the last line converges to
  the $L^2$-norm of the trace of $u^\eps$ on $\Gamma_0$. We have
  obtained that the sequence $U^\eps|_{Y_U}$ is uniformly bounded in
  $L^2(Y_U)$.

  Regarding $U^\eps|_{Y_V}$ we perform the same calculation and use,
  in the last step,
  \begin{align*}
    &\int_{Y_V}|U^\eps(y)|^2\,dy
    \leq C \frac1\eps\int_c^d \int_{L\eps}^{(L+V)\eps} |u^\eps(x)|^2\,dx\\
    &\qquad \leq 2C \frac1\eps\int_c^d \int_{L\eps}^{(L+V)\eps} |u^\eps(x) - v^\eps(x_1)|^2\,dx
      + 2C \frac1\eps\int_c^d \int_{L\eps}^{(L+V)\eps} |v^\eps(x_1)|^2\,dx\\
    &\qquad \leq O(\eps) + 2C V \int_c^d |v^\eps(x_1)|^2\,dx_1 \le C\,,
  \end{align*}
  where we used the one-dimensional Poincar\'e (also called
  Poincar\'e-Wirtinger) inequality with averages $v^\eps(x_1)$ for the
  first integral, exploiting that the domain size is $V\eps$. In the
  last estimate we used the boundedness of $v^\eps$ in $L^2((0,a))$
  that was assumed in Assumption \ref {ass:v-w}.

  The above estimates allow to proceed with the weak $L^2$-compactness
  of bounded sequences.  There exist limit functions $U$ and $V$ such
  that, up to a subsequence, $U^\eps|_{Y_U}\weakto U$ in
  $L^2\left(Y_U\right)$ and $U^\eps|_{Y_V}\weak V$ in
  $L^2\left(Y_V\right)$ as $\eps\to 0$. It is not difficult to verify
  that the limit functions $U$ and $V$ are constant functions. Indeed,
  the gradient of $U^\eps|_{Y_U}$ satisfies, in the limit $\eps\to 0$,
  \begin{align*}
    \int_{Y_U}|\nabla U^\eps(y)|^2\,dy
    &\leq\frac{\eps^2}{|\calK_\eps|}\sum_{k_1\in \calK_\eps} \int_{Y_U}|\nabla
                   u^\eps(\eps(y_1+k_1),\eps y_2)|^2\,dy\\
    &= \frac{1}{|\calK_\eps|}\sum_{k_1\in
         \calK_\eps}\int_{\left(k_1\eps,k_1\eps + \eps\right)\times
          \left(-\eps,0\right)}|\nabla u^\eps(x)|^2\,dx\\
    &\leq  \left(\frac{\eps}{d-c}+O(\eps^2)\right)\int_{(c,d)\times
        (-\eps,0)}|\nabla u^\eps(x)|^2\,dx\to 0\,,
  \end{align*}
  since $u^\eps$ is bounded in $H^1(\Omega_\eps)$. Analogously,
  $\|\nabla U^\eps\|_{L^2\left(Y_V\right)}\to 0$ and we obtain
  $\nabla U = \nabla V = 0$. As a consequence, for two real numbers
  $\xi_U, \xi_V \in \R$, the constant functions are $U\equiv \xi_U$
  and $V\equiv \xi_V$.

  In our next step we identify the constants $\xi_U$ and
  $\xi_V$. There holds
  \begin{align*}
    \xi_U\leftarrow
    &\int_{Y_U}U^\eps(y)\,dy=\frac{1}{|\calK_\eps|}\sum_{k_1\in \calK_\eps}
      \int_{Y_U} u^\eps(\eps(y_1+k_1,\eps y_2)\,dy\\
    =&\frac{1}{\eps^2|\calK_\eps|}\sum_{k_1\in \calK_\eps}
       \int_{\left(k_1\eps,k_1\eps + \eps\right)\times \left(-\eps,0\right)}u^\eps(x_1,x_2)\,dx\\
    =&\left(\frac{1}{d-c}+O(\eps)\right)\left(\frac{1}{\eps}
       \int_{I^\eps_{(c,d)}\times(-\eps,0)}u^\eps(x_1,x_2)\,dx\right)\\
    \rightarrow\,&\frac{1}{d-c}\int_{c}^d u(x_1,0)\,dx_1\,.
  \end{align*}
  Analogously, using definition \eqref{eq:v-eps} of $v^\eps$ and the
  weak convergence $v_\eps\weakto v$ in $L^2(I)$,
  \begin{align*}
    V\, \xi_V
    &\leftarrow \int_{Y_V}U^\eps(y)\,dy=\frac{1}{|\calK_\eps|}\sum_{k_1\in \calK_\eps}
      \int_{Y_V} u^\eps(\eps(y_1+k_1,\eps y_2)\,dy\\
    &=\left(\frac{1}{d-c}+O(\eps)\right)V\left(\frac{1}{V\eps}
      \int_{I^\eps_{(c,d)}\times(\eps L,\eps(L+V))}u^\eps(x)\,dx\right)\\
    &=\left(\frac{V}{d-c}+O(\eps)\right)\int_{I^\eps_{(c,d)}}v^\eps(x_1)\,dx_1\\
    &\rightarrow \frac{V}{d-c}\int_c^dv(x_1)\,dx_1\,.
  \end{align*}
  We have found
  \begin{equation}
    \label{eq:identifyxi}
    \xi_U = \frac{1}{d-c}\int_{c}^d u(x_1,0)\,dx_1\quad\text{and}\quad
    \xi_V = \frac{1}{d-c}\int_c^d v(x_1)\,dx_1\,.
  \end{equation}

  At this point, we identified the averages of $U^\eps$ (below and
  above the channel) with $u$ and $v$. In order to check \eqref
  {eq:meanvalueschannels1}--\eqref {eq:meanvalueschannels2}, it
  remains to relate averages of $U^\eps$ in the bulk areas to averages
  of $U^\eps$ in the ends of the channel. This can be done with a
  Lemma that was proved and used in \cite
  {Lamacz-Schweizer-manyres-2017}.

  We use Lemma A.1 of \cite {Lamacz-Schweizer-manyres-2017} with
  slightly adapted notation. The obstacle in the single cell $Y$ is
  given by
  $\Sigma_Y^\eps := Y\setminus (\overline{Y_U} \cup\overline{Y_V}
  \cup\overline{C_Y^\eps})$. We furthermore assume only boundedness of
  the gradients of $U^\eps$ and not the $L^2$-boundedness everywhere
  (also in the channel). An inspection of the proof in \cite
  {Lamacz-Schweizer-manyres-2017} shows that this is sufficient.

  The essential part of the proof is the following: With the
  tangential vector in each channel being $e_2$, one considers the
  functions $V^\eps := \del_2 U^\eps$. These functions solve the same
  Helmholtz equation and they satisfy homogeneous boundary conditions:
  Dirichlet conditions on one part of the boundary, Neumann conditions
  on the other. This allows to multiply the equation for $V^\eps$ with
  $V^\eps$. One finds uniform $H^1$-estimates for $V^\eps$ which yield
  uniform $H^2$-estimates for $U^\eps$.  The embedding
  $H^2(Y_U) \subset C^0(Y_U)$ (accordingly for $Y_V$) allows to
  compare point values of $U^\eps$ with averages of $U^\eps$.

  \begin{lemma}[Adaption of Lemma A.1 from \cite
    {Lamacz-Schweizer-manyres-2017}]
    Let $U^\eps:Y\setminus \overline{\Sigma_Y^\eps}\to\R$ be a family
    of $H^1$-functions such that the $L^2$-norms of $\nabla U^\eps$
    are bounded. We assume that every $U^\eps$ solves the Helmholtz
    equation
    \begin{align*}
      -\Delta U^\eps&=\omega^2\eps^2 U^\eps\quad
                      \text{in } Y\setminus \overline{\Sigma_Y^\eps}\,,\\
      \del_n U^\eps&=0\quad\text{on } \del\Sigma_Y^\eps\,.
    \end{align*}
    No boundary conditions are imposed on $\del Y$. Assume that
    \begin{align*}
      &U^\eps|_{Y_U}\weakto\xi_U\quad\text{in }L^2(Y_U)\,,\\
      &U^\eps|_{Y_V}\weakto\xi_V\quad\text{in }L^2(Y_V)\,,
    \end{align*}
    as $\eps\to 0$. Then
    \begin{align}
      \label{eq:resultA1}
      \mean_{\Gamma_Y^{\eps,U}}U^\eps(y)\,d\mathcal{H}^1(y)\to \xi_U\quad\text{and}
      \quad \mean_{\Gamma_Y^{\eps,V}}V_\eps(y)\,d\mathcal{H}^1(y)\to \xi_V\,.
    \end{align}
  \end{lemma}

  The Lemma can indeed be applied. (a) $U^\eps$ solves the (rescaled)
  Helmholtz equation with Neumann boundary condition since $u^\eps$
  satisfies the (non-rescaled) system.  (b) The $L^2$-boundedness of
  $\nabla U^\eps$ follows easily from $H^1$-boundedness of $u^\eps$
  (compare the calculations above in this proof). (c) The weak $L^2$
  limits $\xi_U$ and $\xi_V$ have been verified above.

  \smallskip With the result of the lemma at hand, it only remains to
  compare the limits in \eqref {eq:meanvalueschannels1}--\eqref
  {eq:meanvalueschannels2} with the limits in \eqref{eq:resultA1}. We
  calculate with
  $\Gamma_Y^{\eps,U}:=\overline{C^\eps_Y}\cap
  \overline{Y_U}=[0,\alpha\eps^2]\times\{0\}$:
  \begin{align*}
    &\mean_{\Gamma_Y^{\eps,U}}U^\eps(y)\,d\mathcal{H}^1(y)
    = \frac{1}{\alpha\eps^2} \int_{\Gamma_Y^{\eps,U}}\frac{1}{|\calK_\eps|}
      \sum_{k_1\in \calK_\eps}u^\eps(\eps(y_1+k_1,\eps y_2)\,d\mathcal{H}^1(y)\\
    &\qquad
      =\frac{1}{\alpha\eps^2}\frac{1}{|\calK_\eps|}
      \frac1\eps\int_{\Gamma_\eps^{U}}u^\eps(x)\,d\mathcal{H}^1(x)
      =\left(\frac{1}{\eps^2\alpha(d-c)} + O\left(\frac{1}{\eps}\right)\right)
      \int_{\Gamma_\eps^{U}}u^\eps(x)\,d\mathcal{H}^1(x)\,.
  \end{align*}
  The results \eqref{eq:resultA1} and \eqref{eq:identifyxi} thus imply
  \begin{align*}
    \frac{1}{\eps^2}\int_{\Gamma_\eps^U}u^\eps(x)\,d\mathcal{H}^1(x)
    &\to \alpha (d-c) \xi_U = \alpha \int_c^d u(x_1,0)\,dx_1\,,
  \end{align*}
  which is the claim \eqref{eq:meanvalueschannels2}. The limit
  \eqref{eq:meanvalueschannels1} is obtained in an analogous way.
\end{proof}

It remains to derive a further relation between $u$, $v$, and $j$. We
obtain a relation from mass conservation in the resonator volume: The
flux through the channels (and hence the density $j$) can be expressed
in terms of $v$.

\begin{proposition}[Mass conservation]
  \label{prop:B}
  Let $j$ be as in Proposition \ref{prop:geometricflowrule} and $v$ as
  in Assumption \ref{ass:v-w}. Then
  \begin{equation}
    \label{eq:densityjstar-distribution}
    j = V \del_1^2 v + V \omega^2\, v
  \end{equation}
  in the sense of distributions. Furthermore, for every
  $\psi\in C^1(\bar I)$,
  \begin{equation}
    \label{eq:densityjstar1}
    \int_I j(x_1) \psi(x_1)\, dx_1 =
    - \int_I V \del_1 v(x_1) \del_1 \psi(x_1)\, dx_1
    + \int_I V \omega^2 v(x_1) \psi(x_1) \, dx_1\,,
  \end{equation}
  and $v$ has the regularity $\del^2_1 v\in L^1(I)$.
\end{proposition}

\begin{proof}
  We fix $\psi\in C^2(\bar I) = C^2([0,a])$ with
  $\del_1 \psi(0) = \del_1 \psi(a) = 0$. We use $\theta^\eps$ of
  \eqref{eq:functionlargegradient} and consider
  $\fhi^\eps(x_1,x_2) := \psi(x_1) \theta^\eps(x_2)$ in equation
  \eqref{eq:weakformweps} for $w^\eps$.  The left hand side vanishes
  since $\fhi^\eps$ vanishes in $\Omega_0$. There remains
  \begin{equation}
    \label{eq:weakformsimple}
    \frac{1}{\eps} \int_{S_\eps \cup C_\eps} \nabla u^\eps\cdot \nabla\fhi^\eps
    =\frac{\omega^2}{\eps} \int_{S_\eps \cup C_\eps}  u^\eps\, \fhi^\eps\,.
  \end{equation}

  {\em Left hand side of \eqref{eq:weakformsimple}.} We calculate
  \begin{align*}
    \frac{1}{\eps} \int_{S_\eps \cup C_\eps} \nabla u^\eps\cdot \nabla\fhi^\eps
    &= \frac{1}{\eps} \int_{ C_\eps} \del_1 u^\eps(x_1,x_2) \del_1\psi(x_1)\,\frac{x_2}{\eps L}
      + \frac{1}{\eps} \int_{S_\eps} \del_1 u^\eps(x_1,x_2) \del_1\psi(x_1)\\
    &\quad + \frac{1}{\eps} \int_{ C_\eps}\del_2 u^\eps\frac{1}{L\eps}\psi(x_1)\,.
  \end{align*}
  Since $\del_1 u^\eps$ satisfies \eqref{eq:assdel1ueps}, the first
  integral vanishes in the limit $\eps\to 0$. For the second term we
  find
  \begin{equation*}
    \frac{1}{\eps} \int_{S_\eps} \del_1 u^\eps(x_1,x_2) \del_1\psi(x_1)
    =-\frac{1}{\eps}\int_{S_\eps} u^\eps(x_1,x_2) \del^2_1\psi(x_1)
    \to- V\int_I v(x_1)\del^2_1\psi(x_1)\,dx_1
  \end{equation*}
  by the weak convergence $v^\eps \weakto v$ in $L^1(I)$.  For the
  last term we exploit Lemma \ref{lem:currentintroduction}, which
  ensures that
  \begin{equation*}
    \frac{1}{\eps} \int_{ C_\eps}\del_2 u^\eps\frac{1}{L\eps}\psi(x_1)
    =\int_{\R^2}j^\eps\psi(x_1)\to \int_{\Gamma_0}\psi(x_1)\,dj_*(x)
    =\int_I \psi(x_1)j(x_1)\,dx_1\,.
  \end{equation*}

  {\em Right hand side of \eqref{eq:weakformsimple}.} We obtain
  \begin{align*}
    \frac{\omega^2}{\eps} \int_{S_\eps \cup C_\eps}  u^\eps\, \fhi^\eps
    &=\frac{\omega^2}{\eps}
      \left(\int_{C_\eps}  u^\eps(x_1,x_2)\, \psi(x_1)\frac{x_2}{\eps L} +
      \int_{S_\eps}  u^\eps(x_1,x_2)\, \psi(x_1)\right)\\
    &\to V\omega^2\int_{I}v(x_1)\psi(x_1)
  \end{align*}
  as $\eps\to 0$, where we used the convergence of averages in
  $S_\eps$ to $v$ and, for the first term,
  $\eps^{-1} \int_{C_\eps} | u^\eps | \le \eps^{-1} \| u^\eps \|_{L^2}
  |C_\eps|^{1/2} \le C \eps^{1/2} \to 0$.  We obtain from
  \eqref{eq:weakformsimple}
  \begin{equation}
    \label{eq:densityjstar1-267}
    \int_I j(x_1) \psi(x_1)\, dx_1 =
     \int_I V v(x_1) \del_1^2 \psi(x_1)\, dx_1
    + \int_I V \omega^2 v(x_1) \psi(x_1) \, dx_1\,.
  \end{equation}

  Relation \eqref {eq:densityjstar1-267} provides \eqref
  {eq:densityjstar-distribution}. In particular, the distribution
  $\del_1^2 v$ is expressed by the $L^1$-functions $v$ and $j$
  (compare Proposition \ref{prop:geometricflowrule}). We therefore
  find $v\in W^{2,1}(I)$.

  Relation \eqref {eq:densityjstar1} follows with another integration
  by parts from \eqref {eq:densityjstar1-267}. The set of test
  functions is dense in $H^1(I)$, hence \eqref {eq:densityjstar1}
  holds for all $\psi\in H^1(I)$ and, in particular, for all
  $\psi\in C^1(\bar I)$.
\end{proof}

We can now formally conclude the proof of our main theorem.

\begin{proof}[Proof of Theorem \ref{thm:mainTheorem}.] Equation \eqref
  {eq:main-w-weak} for $w$ was checked in Proposition \ref{prop:A}.
  Due to Propositions \ref{prop:B} and \ref{prop:geometricflowrule} we
  find
  \begin{align*}
    V(\del^2_1 v(x_1)+\omega^2 v(x_1))=j(x_1)=\frac{\alpha}{L}(v(x_1)-u(x_1,0))
  \end{align*}
  in the sense of distributions.  Re-ordering terms, we may write this
  relation equivalently as
  \begin{align*}
    -\del_1^2 v(x_1)+\left(\frac{\alpha}{LV}-\omega^2\right)v(x_1)
      =\frac{\alpha}{LV}u(x_1,0)\,,
  \end{align*}
  which is relation \eqref{eq:limiteq-v}.  For a test function
  $\psi\in C^1(\bar I)$, the distribution $\del_1^2 v$ can be
  integrated by parts once without boundary terms. We therefore have
  obtained also \eqref{eq:limiteq-v-weak}, which encodes additionally
  the homogeneous Neumann boundary condition for $v$.
\end{proof}

\subsection*{Acknowledgement} This work was initiated when the three
authors attended in summer 2019 workshop 1931 ``Computational
Multiscale Methods'' in Oberwolfach. The invitation and the
hospitality of the institute are gratefully acknowledged.

\bibliographystyle{abbrv}
\bibliography{lit-Helmholtz}

\end{document}


TITLE: Proof that bilinear form in $H_0^1$ is coercive
QUESTION [0 upvotes]: Let $$B(u,v)=\int_I uv + \int_I u'v'$$ where $u,v\in H_0^1(I)$ for a given interval $I=[a,b]\subset\mathbb{R}$.
How can I prove that the bilinear form $B$ is coercive, i.e., that
$$B(u,u)\ge C\Vert u\Vert_{H_0^1(I)}^2\ \ ?
$$
Here, $\Vert u\Vert_{H_0^1(I)}:=\int_I u^2 +\int_I (u')^2$.
I have tried using that this norm is equivalent to $\Vert u\Vert_{L^2}+\Vert u'\Vert_{L^2}$.

REPLY [1 votes]: Your bilinear form is $B:H_0^1(I)\times H_0^1(I)\to\mathbb{R}$ given by
$$B(u,v)=\int_I uv + \int_I u'v'.$$
You want to prove that $B$ is coercive. In other words, you want to prove that there is $C>0$ such that
$$|B(u,u)|\geq C\|u\|_{H_0^1(I)}^2.\tag{1}$$
To do this, you have to specify what is the norm in $H_0^1(I)$ (I think that there is a typo in your post). Well, there are three usual possibilities:
$\displaystyle\|u\|_{H_0^1(I)}=\left(\int_Iu^2+\int_I(u')^2\right)^{1/2}$ which can be written as $\displaystyle\|u\|_{H_0^1(I)}=\sqrt{\|u\|_{L^2(I)}^2+\|u'\|_{L^2(I)}^2}$;
$\displaystyle\|u\|_{H_0^1(I)}=\left(\int_Iu^2\right)^{1/2}+\left(\int_I(u')^2\right)^{1/2}$ which can be written as $\displaystyle\|u\|_{H_0^1(I)}=\|u\|_{L^2(I)}+\|u'\|_{L^2(I)}$;
$\displaystyle\|u\|_{H_0^1(I)}=\int_I(u')^2$ which can be written as $\displaystyle\|u\|_{H_0^1(I)}=\|u'\|_{L^2(I)}$.
These three norms are all equivalents and thus you can use any of them to prove the desired inequality $(1)$.
Well, using the first possibility above, we have
$$B(u,u)=\int_I u^2 + \int_I (u')^2=\|u\|_{H_0^1(I)}^2$$
which imply $(1)$ with $C=1$ (and thus the the equivalence that you said is not needed).


text:
\begin{document}
\setlength{\evensidemargin}{.1in}
\setlength{\oddsidemargin}{.1in}
\maketitle
\begin{abstract}
We show that, in finite dimensions, the only monotone metrics on
the space of invertible density matrices for which the $(+1)$ and $(-1)$
affine connections are mutually dual are constant multiples of the
Bogoliubov-Kubo-Mori metric.
\end{abstract}

\section{Introduction}

The first task of quantum information geometry is to extend to
non-commutative probability spaces the uniqueness and rigidity of
the geometrical structures used in the classical version of the
theory. This includes, for instance, the uniqueness result of Chentsov concerning
the Fisher metric \cite{Chentsov} and the two equivalent
definitions of $\alpha$-connections given by Amari \cite{Amari1}:
either using an $\alpha$-embedding of the form $p \mapsto
\frac{2}{1-\alpha}p^{\frac{1-\alpha}{2}}$ or as the convex mixture
of the exponential and mixture connections.

As for the first problem, Chentsov himself made the first attempt
to find the possible Riemannian metrics on a quantum information
manifold with the property of having its line element reduced
under stochastic maps \cite{MorozovaChentsov} (cf \cite{Petz3}). This monotonicity
property, which is the quantum analogue of being reduced under Markov morphisms,
was later investigated by Petz \cite{Petz3}. Unlike the
classical case, he found that there are infinitely many Riemannian metrics
satisfying it. Concerning the second problem, several definitions
of $\alpha$-connections have been proposed
\cite{Nagaoka,Hasegawa1,GibiliscoIsola}, both for finite
and infinite dimensional quantum systems. Since some of these
definitions involve finding  dual connections with respect to
some chosen Fisher metric, it is clear that the multitude
of possible candidates for the metric encourages the appearance of
non-equivalent definitions for the $\alpha$-connections.

We take the position that quantum information manifolds are
equipped with two natural flat connections: the mixture
connection, obtained from the linear structure of trace class
operators themselves, and the exponential connection, obtained
when combinations of states are performed by adding their
logarithms \cite{GrasselliStreater,Streater}. In \S2 we present our
preferred definition of the
$\pm 1$-connections in finite dimensional quantum information manifolds.

Following Amari \cite{Amari1,Amari2}, we consider duality as a fundamental structure
to be explored. Thus, given these two connections, we should ask what are the Riemannian
metrics that make them dual. We give our answer to this question in \S3.
In \S4, we combine this result with Petz's characterisation
of monotone metrics in finite dimensional quantum systems to
find that the {\em BKM} metric \cite{Bogoliubov,Kubo,Mori} is, up to a
factor, the unique monotone Riemannian metric with respect to
which the exponential and mixture connections are dual.

\section{The Exponential and Mixture Connections}

Let ${\cal H}^N$ be a finite dimensional complex Hilbert space, $\cal A$
the subspace of self-adjoint operators and $\cal M$ the set of all
invertible density operators on ${\cal H}^N$. Then $\cal A$ is an
$N^2$-dimensional real vector space and $\cal M$  is
an $n$-dimensional submanifold with $n=N^2-1$. Defining
the $1$-embedding of $\cal M$ into $\cal A$ as
\begin{eqnarray*}
\ell_1 & : & {\cal M} \rightarrow{\cal A}\\
& & \rho \mapsto \log \rho,
\end{eqnarray*}
we can use the linear space structure of $\cal A$ to obtain a representation
of the tangent bundle of $\cal M$ in terms of operators in $\cal A$ even
though $\cal M$ is not a vector space itself. At each point $\rho \in {\cal M}$,
consider the subspace ${\cal A}_{\rho}=\{A \in {\cal A} : \mbox{Tr} (\rho A) =
0 \}$ of $\cal A$, called the space of `scores'; we define the isomorphism
\begin{eqnarray*}
(\ell_1)_{*(\rho)} & : & T_{\rho}{\cal M} \rightarrow {\cal A}_{\rho} \\
& & v \mapsto (\ell_1 \circ \gamma)^\prime (0),
\end{eqnarray*}
where $\gamma :(-\varepsilon,\varepsilon) \rightarrow {\cal M}$ is a
curve in the equivalence class of the tangent vector $v$. We call this
isomorphism the
$1$-representation of the tangent space $T_{\rho}{\cal M}$. If
$(\theta^1,\ldots ,\theta^n)$ is a coordinate system for $\cal M$,
put $\partial_i:=\partial/\partial\theta^i,\;i=1,\ldots,n$, evaluated
at $\rho$. Then the $1$-representation of the basis $\left\{\partial_1,
\ldots ,{\partial_n}\right\}$ of $T_{\rho}{\cal M}$ is
$\left\{ \frac{\partial\log\rho}{\partial\theta^1},
\ldots ,\frac{\partial\log\rho}{\partial\theta^n}\right\}$. The
$1$-representation of a vector field $X$ on $\cal M$ is therefore
the $\cal A$-valued function $(X)^{(1)}$ given by
$(X)^{(1)}(\rho)=(\ell_1)_{*(\rho)}X_{\rho}$.

The exponential or $1$-connection is the connection obtained
from the $1$-embedding through the following parallel transport
\cite{Streater}
\begin{eqnarray*}
\tau^{(1)}_{\rho_0,\rho_1} & : & T_{\rho_0}{\cal M} \rightarrow T_{\rho_1}{\cal M}
\\                   &   & v \mapsto
(\ell_1)_{*(\rho_1)}^{-1}\left((\ell_1)_{*(\rho_0)}v -
\mbox{Tr}[\rho_1(\ell_1)_{*(\rho_0)}v]\right).
\end{eqnarray*}
Giving the parallel transport in a neighbourhood of $\rho$ is equivalent
to specifying the
covariant derivative. It is readily verified that the $1$-representation of
the $1$-covariant derivative, applied to the vector field $\partial_j:=
\partial/\partial\theta^j$, is
\begin{equation}
\label{exp-cov}
\left(\nabla^{(1)}_{\partial_i}\frac{\partial}
{\partial\theta^j}\right)^{(1)}=
\frac{\partial^2\log\rho}{\partial\theta^i\partial\theta^j}-
\mbox{Tr}\left(\rho\frac{\partial^2\log\rho}{\partial\theta^i\partial
\theta^j}\right)
\end{equation}

At a more abstract level, the construction above corresponds to making
${\cal M}$ into an affine space and endowing it with the {\em natural} flat
connection \cite{Murray}. Rather than exploring this
concept further, we benefit from dealing with finite dimensional
spaces and prove that the $1$-connection is flat by exhibiting an affine
coordinate system for it. Let $\left\{{\bf{1}}, X_1, \ldots, X_n\right\}$ be
a basis for $\cal A$. Since $\log\rho \in {\cal A}$, there
exist real numbers $\left\{\theta^1, \ldots, \theta^n, \Psi \right\}$
such that \[\log\rho = \theta^1X_1 + \cdots + \theta^nX_n -
\Psi{\bf 1},\] that is,
\begin{equation}
\rho=\exp\left(\theta^1X_1 + \cdots + \theta^nX_n -
\Psi{\bf 1}\right).
\end{equation}
The normalisation condition $\mbox{Tr}\rho=1$
means, however, that only $n$ among these numbers are independent,
so we choose $\Psi\equiv\Psi(\theta)$ to be the one determined by
the others. Then $\theta = (\theta^1, \ldots, \theta^n)$ form
a $1$-affine coordinate system, as can be seen from the
following calculation
\begin{eqnarray*}
\left(\nabla^{(1)}_{\partial_i}\frac{\partial}{\partial\theta^j}\right)^{(1)}
& = & \frac{\partial^2\log\rho}{\partial\theta^i\partial\theta^j}-
\mbox{Tr}\left(\rho\frac{\partial^2\log\rho}{\partial\theta^i\partial\theta^j}\right)\\
& = & -\frac{\partial^2\Psi}{\partial\theta^i\partial\theta^j}(\theta)
+\mbox{Tr}\left(\rho\frac{\partial^2\Psi}{\partial\theta^i\partial\theta^j}(\theta)\right)\\
& = & 0.
\end{eqnarray*}
Now let ${\cal A}_0$ be the subspace of traceless operators in $\cal A$.
Consider the $-1$-embedding
\begin{eqnarray*}
\ell_{-1} & : & {\cal M} \rightarrow {\cal A} \\
       &   & \rho \mapsto \rho,
\end{eqnarray*}
and define, at each $\rho \in {\cal M}$, the $-1$-representation
of tangent vectors as
\begin{eqnarray*}
(\ell_{-1})_{*(\rho)} & : & T_{\rho}{\cal M} \rightarrow {\cal A}_0 \\
             &   & v \mapsto (\ell_{-1} \circ \gamma)^\prime(0),
\end{eqnarray*}
where $\gamma :(-\varepsilon,\varepsilon)\rightarrow{\cal M}$ is again a
curve in the equivalence class of the tangent vector $v$.
In coordinates, the $-1$-representation of the basis
$\left\{\partial_1,\ldots ,\partial_n\right\}$
of $T_{\rho}{\cal M}$ is $\left\{ \frac{\partial\rho}{\partial\theta^1},
\ldots ,\frac{\partial\rho}{\partial\theta^n}\right\}$. As before, the
$-1$-representation of a vector field $X$ on $\cal M$ is an
${\cal A}_0$-valued function denoted by $(X)^{(-1)}$ or $X^-$.

We obtain the mixture or $-1$-connection by defining the parallel transport
\begin{eqnarray*}
\tau^{(-1)}_{\rho_0,\rho_1} & : & T_{\rho_0}{\cal M} \rightarrow T_{\rho_1}{\cal M}
\\                   &   & v \mapsto
(\ell_{-1})_{*(\rho_1)}^{-1}\left((\ell_{-1})_{*(\rho_0)}v\right),
\end{eqnarray*}
and we find that its covariant derivative in the direction $\partial_i$ is
\begin{equation}
\left(\nabla^{(-1)}_{\partial_i}\frac{\partial}{\partial\theta^j}\right)^{(-1)}=
\frac{\partial^2\rho}{\partial\theta^i\partial\theta^j}.
\end{equation}
If we equip $\cal A$ with the trace norm, then the
$-1$-embedding maps $\cal M$ into the unit sphere $\cal S$ of $\cal A$,
and the $-1$-connection given here is nothing but the projection
onto $\cal S$ of the {\em natural} flat connection in this space. It
turns out that the unit sphere with respect to the trace norm is flat
in  $\cal A$, hence the $-1$-connection is flat on $\cal M$.
Again, it is a convenience of the finite dimensional case that we
can prove flatness of the $-1$-connection by direct construction of a
$-1$-affine coordinate system. Suppose $\left\{X_1, \ldots, X_{n+1}\right\}$
is a normalised basis for $\cal A$, then there exist real numbers $(\xi_1, \ldots, \xi_{n+1})$
such that
\begin{equation}
\rho = \xi_1X_1 + \cdots + \xi_{n+1}X_{n+1}.
\end{equation}
Since $\mbox{Tr}\rho =1$, we can take $\xi_{n+1}= 1 - (\xi_1 + \cdots + \xi_n)$
as a function of the first $n$ independent parameters $\xi_1, \ldots, \xi_{n}$. Then
$\xi = (\xi_1, \ldots, \xi_{n})$ is a $-1$-affine coordinate system for $\cal M$, because
\begin{eqnarray*}
\left(\nabla^{(-1)}_{\frac{\partial}{\partial\xi_i}}\frac{\partial}{\partial\xi_j}\right)^{(-1)}
& = & \frac{\partial^2\rho}{\partial\xi_i\partial\xi_j}\\
& = & \frac{\partial}{\partial\xi_i}(X_j - X_{n+1})\\
& = & 0.
\end{eqnarray*}

\section{Duality and the {\em BKM} Metric}

Two connections $\nabla$ and $\nabla^*$ on a Riemannian manifold $({\cal M},g)$
are dual with respect to $g$ if and only if
\begin{equation}
X g(Y,Z)= g \left(\nabla_X Y,Z \right)+g \left(Y,\nabla^*_X Z\right),
\end{equation}
for any vector fields $X,Y,Z$ on $\cal M$ \cite{Amari1,Murray}. Equivalently,
if $\tau_{\gamma(t)}$ and $\tau^*_{\gamma(t)}$ are the respective parallel
transports along a curve $\{\gamma(t)\}_{0\leq t\leq 1}$ on $\cal M$, with
$\gamma(0)=\rho$, then $\nabla$ and $\nabla^*$ are dual with respect to $g$
if and only if for all $t\in[0,1]$,
\begin{equation}
g_\rho(Y,Z)=g_{\gamma(t)}\left(\tau_{\gamma(t)}Y,\tau^*_{\gamma(t)}Z\right).
\end{equation}

Given any connection $\nabla$ on $({\cal M},g)$, we can always
find a unique connection $\nabla^*$ such that $\nabla$ and
$\nabla^*$ are dual with respect to $g$. On the other hand, given
two connections $\nabla$ and $\nabla^*$, we can ask what are the
possible Riemannian metrics $g$ with respect to which they are
dual. In particular, we want to explore this question for the case
of the exponential and mixture connections on a manifold of
density matrices.

A different concept of duality also used by Amari \cite{Amari1}
is that of dual coordinate systems, regardless of any connection.
Two coordinate systems $\theta = (\theta^i)$ and $\eta = (\eta_i)$ on a Riemannian manifold
$({\cal M},g)$ are dual with respect to $g$ if and only if their
natural bases for $T_\rho{\cal M}$ are {\em biorthogonal} at every point
$\rho\in {\cal M}$, that is,
\[g\left(\frac{\partial}{\partial\theta^i},\frac{\partial}{\partial\eta_j}\right)=\delta^i_j.\]
Equivalently, $\theta = (\theta^i)$ and $\eta = (\eta_i)$ are dual
with respect to $g$ if and only if
\[g_{ij}=\frac{\partial\eta_i}{\partial\theta^j} \quad \mbox{and}
\quad g^{ij}=\frac{\partial\theta_i}{\partial\eta^j}\]
at every point $\rho\in {\cal M}$, where, as usual,
$g^{ij}=(g_{ij})^{-1}$.

The following theorem gives a characterisation of dual coordinate
systems in terms of potential functions, thus bringing
convexity theory and the related duality with respect to Legendre transforms
into the discussion.

\begin{theorem}[Amari, 1985]
When a Riemannian manifold $({\cal M},g)$ has a pair of dual coordinate
systems $(\theta,\eta)$, there exist potential functions $\Psi(\theta)$
and $\Phi(\eta)$ such that
\[g_{ij}(\theta)=\frac{\partial^2\Psi(\theta)}{\partial\theta^i\partial\theta^j}
\quad \mbox{and} \quad
g^{ij}=\frac{\partial^2\Phi(\eta)}{\partial\eta_i\partial\eta_j}.\]
Conversely, when either potential function $\Psi$ or $\Phi$ exists
from which the metric is derived by differentiating it twice, there
exist a pair of dual coordinate systems. The dual
coordinate systems and the potential functions are related by the following Legendre
transforms
\[\theta^i=\frac{\partial\Phi(\eta)}{\partial\eta_i},
\quad \eta_i=\frac{\partial\Psi(\theta)}{\partial\theta^i}\]
and
\[\Psi(\theta)+\Phi(\eta)-\theta^i\eta_i = 0\]
\label{potentials}
\end{theorem}

In contrast to the case of dual connections, dual coordinate
systems do not necessarily exist on every Riemannian manifold \cite{Amari1}.
When the additional property of flatness is required,
the following theorem provides a link between the two concepts of duality.
In the sense used in this paper, a connection $\nabla$ on manifold $\cal M$ is
said to be flat if $\cal M$ admits a global $\nabla$-affine
coordinate system. This is equivalent to its curvature and
torsion both being zero.

\begin{theorem}[Amari, 1985]
Suppose that $\nabla$ and $\nabla^*$ are two flat connections on a manifold $\cal M$.
If they are dual with respect to a Riemannian metric $g$ on
$\cal M$, then there exists a pair $(\theta,\eta)$
of dual coordinate systems such that $\theta$ is $\nabla$-affine
and $\eta$ is a $\nabla^*$-affine.
\label{dualities}
\end{theorem}

We now return to our manifold $\cal M$ of density matrices and
consider the problem of finding a unique Riemannian metric for it.
Using either the $1$ or the $-1$ representation of the
tangent bundle $T{\cal M}$, we define a Riemannian metric on $\cal M$
by a smooth assignment of an inner product $\langle\cdot,\cdot\rangle_
{\rho}$ in
${\cal A}\subset B({\cal H}^N)$ for each point $\rho \in {\cal M}$.
The {\em BKM} (Bogoliubov-Kubo-Mori) metric is the Riemannian metric on
$\cal M$ obtained from the {\em BKM} inner product \cite{Petz2}.
If $A^{(1)},B^{(1)}$ and
$A^{(-1)},B^{(-1)}$ are, respectively, the $1$ and $-1$ representations of
$A,B \in T_{\rho}{\cal M}$, then the {\em BKM} metric has the following
equivalent expressions:
\begin{eqnarray}
g^{\scriptscriptstyle B}_{\rho}(A,B) & = & \mbox{Tr}\left(A^{(-1)}
B^{(1)}\right)\nonumber\\
& = & \int_{0}^{1}\mbox{Tr}\left(\rho^{\lambda}A^{(1)}\rho^{1-\lambda}
B^{(1)}\right)d\lambda\nonumber\\
& = & \int_{0}^{\infty}\mbox{Tr}\left(\frac{1}{t+\rho}A^{(-1)}\frac{1}
{t+\rho}B^{(-1)}\right)dt\label{bkm}
\end{eqnarray}

It is well known that the exponential and mixture connections are
dual with respect to the {\em BKM} metric \cite{Nagaoka,Hasegawa2}.
A natural question is whether it is the unique metric with this
property. The next theorem tells us how much uniqueness can be
achieved from duality alone.

\begin{theorem}
If the connections $\nabla^{(1)}$ and $\nabla^{(-1)}$ are dual
with respect to a Riemannian metric $g$ on $\cal M$, then there
exist a constant (independent of $\rho$) $n\times n$ matrix $M$,
such that $(g_{\rho})_{ij}={\displaystyle \sum_{k=1}^{n}}
M_i^k(g^{\scriptscriptstyle B}_{\rho})_{kj}$, in some $1$-affine
coordinate system. \label{unique1}
\end{theorem}
{\em Proof:} Since the two connections are flat, by
theorem~\ref{dualities}, there exist dual coordinate systems $(\theta,\eta)$
such that $\theta$ is $\nabla^{(1)}$-affine and $\eta$ is
$\nabla^{(-1)}$-affine. Thus, applying theorem~\ref{potentials},
there exist a potential function $\Psi(\theta)$ such that
\[g_{ij}(\theta)=\frac{\partial^2\Psi(\theta)}{\partial\theta^i\partial\theta^j}\]
and
\[\eta_i=\frac{\partial\Psi(\theta)}{\partial\theta^i}.\]
On the other hand, since $\theta$ is $\nabla^{(1)}$-affine, there
exist linearly independent operators\\
$\left\{{\bf{1}}, X_1, \ldots, X_n\right\}$ such that
\begin{equation}
\rho=\exp\left(\theta^1X_1 + \cdots + \theta^nX_n -
\tilde{\Psi}(\theta){\bf 1}\right)
\label{rho},
\end{equation}
where
\begin{equation}
\tilde{\Psi}(\theta)=\log\mbox{Tr}\left[\exp\left(\theta^1X_1 + \cdots +
\theta^nX_n\right)\right].
\label{free-energy}
\end{equation}
Without loss of generality, we can assume that the operators $X_1, \ldots, X_n$
are traceless, since if we add multiples of the identity to any $X_j$
in~(\ref{rho}), we can still have the same parameters $\theta$ as coordinates
for the same point $\rho$ just by modifying the function $\tilde{\Psi}$. But any such set
of operators define a $\nabla^{(-1)}$-affine coordinate system through the formula
\[\tilde{\eta}_i=\mbox{Tr}(\rho X_i),\] because the latter are affinely related to the $\xi$ coordinates defined in section 2 (with $X_{n+1}={\bf 1}/n$).
Differentiating $\tilde{\Psi}$ with respect to $\theta^i$ we obtain
\[\frac{\partial\tilde{\Psi}(\theta)}{\partial\theta^i}=\mbox{Tr}(\rho X_i)=
\tilde{\eta}_i.\]
Thus $\tilde{\eta}_i=\frac{\partial\tilde{\Psi}(\theta)}{\partial\theta^i}$ and
$\eta_i=\frac{\partial\Psi(\theta)}{\partial\theta^i}$
are two $\nabla^{(-1)}$-affine coordinate systems, so they must be
related by an affine transformation \cite{Murray}. So there
exist an $n\times n$ matrix $M$ and numbers $(a_1, \ldots, a_n)$
such that
\[\eta_i = \sum_{k=1}^{n}M_i^k\tilde{\eta}_k + a_i,\]
that is,
\[\frac{\partial\Psi(\theta)}{\partial\theta^i} =
\sum_{k=1}^{n}M_i^k\frac{\partial\Psi(\theta)}{\partial\theta^k} +
a_i,\]
and differentiating this equation  with respect to $\theta^j$ gives
\begin{equation}
g_{ij}(\theta)=\frac{\partial^2\Psi(\theta)}{\partial\theta^i\partial\theta^j}
=
\sum_{k=1}^{n}M_i^k
\frac{\partial^2\tilde{\Psi}(\theta)}{\partial\theta^j\partial\theta^k}.
\label{psis}
\end{equation}
But we can calculate the second derivative of $\tilde{\Psi}$ directly
from~(\ref{free-energy}), obtaining
\begin{eqnarray*}
\frac{\partial^2\tilde{\Psi}(\theta)}{\partial\theta^j\partial\theta^k} &
= & \int_{0}^{1}\mbox{Tr}\left(\rho^{\lambda}\frac{\partial\log\rho}{\partial\theta^j}
\rho^{1-\lambda}\frac{\partial\log\rho}{\partial\theta^k}\right)d\lambda\\
& = & g^{\scriptscriptstyle B}_{\rho}
\left(\frac{\partial}{\partial\theta^j},\frac{\partial}{\partial\theta^k}\right)\\
& = & g^{\scriptscriptstyle B}_{jk}(\theta).
\end{eqnarray*}
Inserting this in~(\ref{psis}) completes the
proof.\hspace{\fill}$\Box$

\section{The Condition of Monotonicity}

We have seen in the previous section that requiring duality
between the exponential and mixture connections reduces the set of
possible Riemannian metrics on $\cal M$ to matrix multiples of
the {\em BKM} metric. We now investigate the effect of imposing a
monotonicity property on this set.

If we use the $-1$-representation to define a Riemannian metric $g$ on $\cal M$
by means of the inner product
$\langle\cdot,\cdot\rangle_{\rho}$ in ${\cal A}\subset B({\cal H}^N)$,
then we say that $g$ is {\em monotone} if and only if
\begin{equation}
\left\langle S(A^{(-1)}),S(A^{(-1)})\right\rangle_{S(\rho)} \leq \left\langle
A^{(-1)},A^{(-1)}\right\rangle_{\rho},
\end{equation}
for every $\rho \in {\cal M}$, $A \in T_{\rho}{\cal M}$, and every
completely positive, trace preserving map $S:{\cal A} \rightarrow {\cal
A}$.

In a series of papers \cite{Petz1,Petz3,PetzSudar}, Petz has given
a complete characterisation of monotone metrics on full matrix
spaces in terms of operator monotone functions. The Chentsov
condition, however, is defined for metrics on the space ${\cal M}$
of faithful {\em states}, and must first be extended to ${\cal A}$
before we can use Petz's results. Let $\widehat{\cal M}$ be the
manifold of faithful weights (the positive definite matrices). Let
$g$ be a metric on $T{\cal M}$. We can extend $g$ to ${\cal
A}\simeq T\widehat{\cal M}$ as follows. At $\rho\in{\cal M}$ and
$\hat{A},\hat{B} \in T\widehat{\cal M}_\rho$, put
$\hat{A}^{(-1)}=A_0\rho+A^-$, where $A_0=\mbox{Tr}\,\hat{A}^{(-1)}
\in{\bf R}$ and $\mbox{Tr}\,A^-=0$, and similarly for $B$. Then
put
\begin{equation}
\hat{g}_\rho(\hat{A},\hat{B}):=A_0B_0+g_\rho(A^-,B^-). \label{ext}
\end{equation}
For $g^{\scriptscriptstyle B}$ this extension coincides with that given by
eq.~(\ref{bkm}). More generally, if $g$ is monotone on $T{\cal M}$, then
$\hat{g}$ is monotone on $T\widehat{\cal M}$. For, let $S$ be a
trace-preserving completely positive map on $T\widehat{\cal M}$, and $\rho\in{\cal M}$.
Then $S$ maps $T{\cal M}$ to itself, and
\[
\hat{g}_{_{S\rho}}(S\hat{A},S\hat{A})=A_0^2+g_{_{S\rho}}(SA^-,SA^-)\leq
A_0^2+ g_\rho(A^-,A^-)=\hat{g}_\rho(A,A).\]

For any metric $\hat{g}$ on $T\widehat{\cal M}$, and putting
$\hat{A}^{(-1)}= A_0\rho+A^-$, we define the positive (super)
operator $K_\rho$ on $\cal A$ by
\begin{equation}
\hat{g}_\rho(\hat{A},\hat{B})=\left\langle
\hat{A}^{(-1)},K_\rho\left(\hat{B}^{(-1)}\right)
\right\rangle_{\scriptscriptstyle H \!
S}=\mbox{Tr}\left(\hat{A}^{(-1)}K_\rho
\left(\hat{B}^{(-1)}\right)\right). \label{metric}
\end{equation}
Note that our $K$ is denoted $K^{-1}$ by Petz. He also defines the
(super) operators, $L_\rho X:=\rho X$ and $R_\rho X:=X\rho$, for
$X \in {\cal A}$, which are also positive. Then he proved
\cite{Petz3}

\begin{theorem}[Petz]
A Riemannian metric $g$ on $\cal A$ is monotone if and only if
\[K_{\rho}=\left(R_{\rho}^{1/2}f(L_{\rho}R_{\rho}^{-1})R_{\rho}^{1/2}
\right)^{-1},\]
where $K_{\rho}$ is defined in~(\ref{metric}) and $f:R^{+}\rightarrow R^{+}$
is an operator monotone function satisfying $f(t)=t f(t^{-1}).$
\label{function}
\end{theorem}

In particular, the {\em BKM} metric is monotone and its corresponding operator
monotone function is
\[f^{\scriptscriptstyle B}(t)=\frac{t-1}{\log t}.\]

Combining this characterisation with our theorem~(\ref{unique1}), we
obtain the following improved uniqueness result.

\begin{theorem}
If the connections $\nabla^{(1)}$ and $\nabla^{(-1)}$ are dual with respect
to a monotone Riemannian metric $g$ on $\cal M$, then $g$ is a
{\em scalar} multiple of the BKM metric.
\end{theorem}
{\em Proof:} We first extend the $(N^2-1)\times(N^2-1)$ matrix $M$ of
theorem~(\ref{unique1}) by one row and column, to a matrix
$\hat{M}$, where $\hat{M}_{ij}=M_{ij},\;(1\leq i,j\leq N^2-1)$,
and
\[
\hat{M}_{i,N^2}=\hat{M}_{N^2,i}=\delta_{i,N^2},
(i=1,\ldots,N^2).\] Now, in the coordinates
$\hat{A}^{(-1)}=A_0\rho+A^-$ of eq.~(\ref{ext}), the extension of
$g$ to $\hat{g}$ in the orthogonal direction $\rho$ is the same
for all $g$. In particular,
\[\hat{g}(A_0\rho,B^-)=\hat{g}^{\scriptscriptstyle B}(A_0\rho,B^-)=0;\hspace{.5in}
\hat{g}(A_0\rho,B_0\rho)=\hat{g}^{\scriptscriptstyle B}(A_0\rho,B_0\rho)=A_0B_0.\]
It follows that the equation proved in theorem~(\ref{unique1}), when expressed in the $-1$-affine coordinates $\eta$ (by taking inverses and using
 theorem~\ref{potentials}), gives the following relation between the kernels of $\hat{g}$ and $\hat{g}^{\scriptscriptstyle B}$, say  $K^g$ and
$K^{\scriptscriptstyle B}$ respectively,
\begin{equation}
K^g_{\rho}=\hat{M}^{\scriptstyle t}K^{\scriptscriptstyle B}_{\rho}.
\end{equation}
Therefore, if $f^g$ and $f^{\scriptscriptstyle B}$ are the operator
monotone functions
corresponding respectively to $g$ and $g^{\scriptscriptstyle B}$, from
theorem~\ref{function}, we have
\begin{eqnarray*}
\left(R_{\rho}^{1/2}f^g(L_{\rho}R_{\rho}^{-1})R_{\rho}^{1/2}\right)^{-1}
& = & \hat{M}^{\scriptstyle t}\left(R_{\rho}^{1/2}f^{\scriptscriptstyle B\! K\! M}
(L_{\rho}R_{\rho}^{-1})R_{\rho}^{1/2}\right)^{-1}\\
\left(R_{\rho}^{1/2}f^g(L_{\rho}R_{\rho}^{-1})R_{\rho}^{1/2}\right)\hat{M}^{\scriptstyle t}
& = & \left(R_{\rho}^{1/2}f^{\scriptscriptstyle B\! K\! M}
(L_{\rho}R_{\rho}^{-1})R_{\rho}^{1/2}\right)\\ \hat{M}^{\scriptstyle t} & = &
f^g(L_{\rho}R_{\rho}^{-1})^{-1}f^{\scriptscriptstyle B\! K\! M}
(L_{\rho}R_{\rho}^{-1}),
\end{eqnarray*}
as everything commutes.
Thus, the operator $\hat{M}$ is given as a function of the operator
$L_{\rho}R_\rho^{-1}$, but it is itself independent of the point $\rho$,
so we conclude that it must be a scalar multiple of the identity
operator.\hspace{\fill}$\Box$

\section{Discussion}

We have proved that, in a finite dimensional quantum system, the
{\em BKM} metric is, up to a scalar factor, the unique monotone
Riemannian metric with respect to
which the exponential and mixture connections are dual. It should
be emphasised that we have considered the manifold of {\em all}
invertible density operators. The uniqueness result might not be
true for a submanifold of $\cal M$, such as occurs in parametric estimation
theory.

We should compare our uniqueness theorem with the results of
Hasegawa \cite{Hasegawa3} and Petz  and Hasegawa
\cite{HasegawaPetz,PetzHasegawa}, who obtained a family of metrics
$\{g^{\alpha}\}$, each corresponding to a divergence based on the
corresponding Wigner-Yanase information, as generalised by Dyson.
They show that these metrics are monotone, and that the
$\alpha$-divergence `admits dual connections', in a sense they
specify. This does not contradict our result; the $\pm
1$-connections are not dual relative to $g^{\alpha}$ unless
$\alpha=\pm 1$, which is the {\em BKM} case.

Having fixed the metric, the next step in the development of the theory is to
define the $\alpha$-connections, for $\alpha \in (-1,1)$. We  look for a
definition that
makes $\nabla^{(\alpha)}$ and $\nabla^{(-\alpha)}$ dual and such that
the extended manifold
$\widehat{\cal M}$ is $\alpha$-flat. Only then we can try to find the
quantum analogue of the $\alpha$-divergence. Duality is easily
achieved if we define the $\alpha$-connections as the convex
mixture $\nabla^{(\alpha)}=\frac{1+\alpha}{2}\nabla^{(1)} +
\frac{1-\alpha}{2}\nabla^{(-1)}$.
Flatness, on the other hand, is more apparent if we use the
$\alpha$-embedding. We have not yet been able to prove that the
two definitions are equivalent. All these questions are deferred to
a subsequent paper.

config: arxiv


TITLE: Union of boxes in $\bar{\mathbb{R}}^3$
QUESTION [3 upvotes]: Let $\bar{\mathbb{R}}$ denote the extended real line. Let    $[x_1,x_2]\times[y_1,y_2]\times [z_1,z_2]$ represent a generic "box" in  $\bar{\mathbb{R}}^3$:
$$[x_1,x_2]\times[y_1,y_2]\times [z_1,z_2]\equiv \{(x,y,z) \in \mathbb{R}^3: x_1\leq x \leq x_2,  y_1\leq y \leq y_2, z_1\leq z \leq z_2\}$$
The square brackets can also be replaced with round brackets, depending on whether the extreme points of each interval are included.
Let $\mathcal{A}_1$ be the collection of boxes of the form
$$
(a+b, \infty]\times [-\infty, a]\times [-\infty,b]
$$
for every $(a,b)\in \mathbb{R}^2$.
Let $\mathcal{A}_2$ be the collection of boxes of the form
$$
[-\infty,a+b)\times [a,\infty]\times [b,\infty]
$$
for every $(a,b)\in \mathbb{R}^2$.
Let $\mathcal{A}\equiv \mathcal{A}_1\cup \mathcal{A}_2$.
Consider a box $B$  such that:
(1) $B\notin \mathcal{A}$.
(2) $B$ is not contained in (is not a "sub-box" of) any element of $\mathcal{A}$.
An example of $B$ satisfying 1-2 is
$$
(0.2, \infty]\times [-\infty, 10]\times [-\infty,11]
$$
Or also
$$
[0.2, \infty]\times [-\infty, 10]\times [-\infty,11]
$$
Question: can a box $B$ satisfying 1-2 be contained in (be a sub-box of) the union of some elements of $\mathcal{A}$? The union should not necessarily be a finite union of elements of  $\mathcal{A}$.
Conjecture: I think the answer to my question is no. I don't know how to prove it, though.
Also, I think that the following holds:
$$
\bar{\mathbb{R}}^3\setminus \{\cup_{D\in \mathcal{A}}D\} = \{(x,y,z)\in \mathbb{R}^3: x-y-z=0\}
$$
That is: once we remove from $\bar{\mathbb{R}}^3$ the union of the boxes in $\mathcal{A}$, we get a plane. I'm not sure  how to use this information, though.

REPLY [1 votes]: If you allow arbitrary combinations of open and closed brackets for $B$, but still restrict the boxes in $\mathcal{A}$ to have open and closed brackets as stated above, then your conjecture is false - a counter-example exists due to a technicality.
Claim: $B = [0, 1] \times [-1, 0) \times [-1, 0)$ satisfy 1-2.
$B \notin \mathcal{A}$ is obvious.
Next consider whether $B$ can be a sub-box of some $A \in \mathcal{A}_1$.  Because the $x$-range of $B$ is $[0,1]$, we need $a + b < 0$ since any box $A$ has $x$-range $(a+b, +\infty]$.  However this implies $a < 0$ or $b < 0$, which would violate either $B$'s $y$-range or $z$-range.
Finally consider whether $B$ can be a sub-box of some $A \in \mathcal{A}_2$.  Similarly $B$'s $x$-range requires $a+b > 1$, which implies $a > 0$ or $b> 0$, which violate either $B$'s $y$-range or $z$-range. $\square$
However, $B$ does not contain any point in the plane $P: x - y - z = 0$.  Therefore, $B \subset \bigcup_{D\in \mathcal{A}} D$ since the union is everything but the plane.

The counter-example relies on the fact that, as defined, the boxes in $\mathcal{A}$ have open and closed brackets a certain way, and the counter-example goes a different way.  I have no doubt that if you allow more kinds of boxes in $\mathcal{A}$, while still omitting the plane $P$, then your conjecture should be true i.e. no counter-example can exist.  In fact I think the proof should be conceptually straightforward (but possibly tedious) via a continuity argument.

Added Dec 29: I've been thinking about the new $\mathcal{A}_1, \mathcal{A}_2$ in your comments.  I think the continuity argument would go like something like this: Define function $f(x,y,z) = x - y - z$.  Then it should be easy (but perhaps tedious due to the open/closed brackets case analysis) to show all the following:

$\bigcup_{D\in \mathcal{A}} D$ is the complement of the plane $P: f(x,y,z) = 0$.

Every box $B$ (whether it satisfies 1-2 or not) can be classified into one of the following 3 cases:

If $f(p) > 0$ for every point $p = (x,y,z) \in B$, then $B$ is a subbox of $A \in \mathcal{A}_1$ with $a = y_2, b = z_2$ and appropriate open/closed brackets.

Similarly, if $f(p) < 0$ for every point $p = (x,y,z) \in B$, then $B$ is a subbox of some $A \in \mathcal{A}_2$ with $a = y_1, b = z_1$ and appropriate open/closed brackets.

Finally, if there are points $p, q \in B$ s.t. $f(p) \ge 0, f(q) \le 0$, then by continuity of $f$ there also exists point $r \in B$ s.t. $f(r) = 0$, e.g. by considering $f$ values along the line segment from $p$ to $q$.  But then $r \in P$ and therefore $B$ cannot be covered by $\bigcup_{D\in \mathcal{A}} D$.


TITLE: Hypergeometric functions inequality
QUESTION [4 upvotes]: Let $_2F_1(a,b;c,z)$ be the (Gauss) hypergeometric function, and $m$ and $n$ positive integers.
From a simple plot it looks like
$_2F_1(m+n,1,m+1,\frac{m}{m+n})>\frac{m}{n} \,_2F_1(m+n,1,n+1,\frac{n}{m+n})$
when $m<n$, but how do I prove this?
Hope this is not too trivial, I am not very familiar with such functions.

REPLY [4 votes]: Using Euler's transformation of the Gauss's hypergeometric function:
$$
    {}_2 F_1\left(a,b; c; z\right) = (1-z)^{c-a-b} \cdot {}_2F_1\left(c-a,c-b; c; z\right)
$$
for $b=1$, $a=m+n$ and $c=m+1$ we have a "closed form" expression:
$$ \begin{eqnarray}
   {}_2 F_1\left(m+n,1; m+1; \frac{m}{m+n}\right) &=& \left(\frac{n}{n+m}\right)^{-n} {}_2F_1\left( 1-n, m; m+1; \frac{m}{m+n} \right) \\ &=& m \left(\frac{n+m}{n}\right)^{n} \cdot \sum_{k=0}^{n-1} \frac{1}{k+m} \frac{(1-n)_k}{k!} \left(\frac{m}{n+m}\right)^k \\
 &=& m \left(\frac{n+m}{n}\right)^{n} \cdot \sum_{k=0}^{n-1} \frac{(-1)^k}{k+m} \binom{n-1}{k} \left(\frac{m}{n+m}\right)^k \\
  &=& m \left(\frac{n+m}{n}\right)^{n} \cdot \left(\frac{n+m}{m}\right)^{m} \int_0^{\frac{m}{n+m}} \left(1-x\right)^{n-1} x^{m-1} \mathrm{d} x
\end{eqnarray}
$$
Thus:
$$
  \frac{1}{m} {}_2 F_1\left(m+n,1; m+1; \frac{m}{m+n}\right) -
  \frac{1}{n} {}_2 F_1\left(m+n,1; n+1; \frac{n}{m+n}\right) =
    \frac{(n+m)^{n+m}}{n^n m^m} \left( \int_0^{\frac{m}{n+m}} \left(1-x\right)^{n-1} x^{m-1} \mathrm{d} x - \int_0^{\frac{n}{n+m}} \left(1-x\right)^{m-1} x^{n-1} \mathrm{d} x \right) =
    \frac{(n+m)^{n+m}}{n^n m^m} \left( \int_0^{\frac{m}{n+m}} \left(1-x\right)^{n-1} x^{m-1} \mathrm{d} x - \int_{\frac{m}{n+m}}^1 \left(1-x\right)^{n-1} x^{m-1} \mathrm{d} x \right) = \\
\frac{(n+m)^{n+m}}{n^n m^m}  \int_0^1 \operatorname{sgn}\left({\frac{m}{n+m}}-x\right)\left(1-x\right)^{n-1} x^{m-1} \mathrm{d} x
$$
Interpreting $x$ as a beta random variable the last expression becomes:
$$
  \frac{(n+m)^{n+m}}{\Gamma(n+m)} \frac{n^n}{\Gamma(n)} \frac{m^m}{\Gamma(m)} \mathbb{E}\left(  \operatorname{sgn}\left(\mu-X\right) \right)  =
   \frac{(n+m)^{n+m}}{\Gamma(n+m)} \frac{n^n}{\Gamma(n)} \frac{m^m}{\Gamma(m)} \left( \mathbb{P}\left( X < \mu \right) - \mathbb{P}\left( X > \mu \right)\right)
$$
where $\mu = \mathbb{E}(X) = \frac{m}{m+n}$, where $X$ follows $\mathrm{Be}(m,n)$. Given that $m<n$, random variable $X$ is positively skewed, and the expression is positive.


TITLE: Limit that appears in particle physics calculation
QUESTION [1 upvotes]: I came across the following statement, which I do not see how to prove it:
$$\lim_{\epsilon \to 0} \frac{\epsilon}{x^2+\epsilon^2} =\pi \delta(x)\,.$$
Any ideas how to proceed?

REPLY [1 votes]: The area under the function
$$\int_{-\infty}^\infty\frac{dx}{\pi(1+x^2)}=1$$
is unit.
Then by scaling,
$$\int_{-\infty}^\infty\frac{d\dfrac x\epsilon}{\pi\left(1+\dfrac{x^2}{\epsilon^2}\right)}=\int_{-\infty}^\infty\frac{\epsilon\,dx}{\pi\left(\epsilon^2+x^2\right)}.$$
As $\epsilon$ decreases, the peak gets narrower and narrower.