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# Estimates on the Neumann and Steklov principal eigenvalues of collapsing
domains

P. Acampora, V. Amato, E. Cristoforoni

###### Abstract

We investigate the relationship between the Neumann and Steklov principal
eigenvalues emerging from the study of collapsing convex domains in
$\mathbb{R}^{2}$. Such a relationship allows us to give a partial proof of a
conjecture concerning estimates of the ratio of the former to the latter: we
show that thinning triangles maximize the ratio among convex thinning sets,
while thinning rectangles minimize the ratio among convex thinning with some
symmetry property.
MSC 2020: 35P15, 49Q10, 52A40
Keywords: Neumann eigenvalue, Steklov eigenvalue, thinning convex sets, Sturm-
Liouville

##  1 Introduction

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded, open, connected and Lipschitz
set. We define the Neumann and Steklov eigenvalues as follows: find positive
constants $\mu,\sigma$ such that there exist non-zero solutions to the
boundary value problems

$\begin{dcases}-\Delta u=\mu u&\text{in }\Omega,\\\ \frac{\partial
u}{\partial\nu}=0&\text{on
}\partial\Omega,\end{dcases}\qquad\qquad\qquad\begin{dcases}\Delta
v=0&\text{in }\Omega,\\\ \frac{\partial v}{\partial\nu}=\sigma v&\text{on
}\partial\Omega.\end{dcases}$

The regularity assumption we made on $\Omega$ ensures that we can find two
increasing and divergent sequences of eigenvalues

$\begin{split}0=\mu_{0}(\Omega)<\mu_{1}(\Omega)\leq\mu_{2}(\Omega)\leq\dots\leq\mu_{k}(\Omega)\leq\dots,\\\\[5.0pt]
0=\sigma_{0}(\Omega)<\sigma_{1}(\Omega)\leq\sigma_{2}(\Omega)\leq\dots\leq\sigma_{k}(\Omega)\leq\dots,\end{split}$

which are the spectrum of the Neumann laplacian and the spectrum of the
Dirichlet-to-Neumann map respectively. We recall the variational
characterization of the eigenvalues, for $k\geq 0$:

$\mu_{k}(\Omega)=\inf_{E\in\mathcal{S}_{k+1}(\Omega)}\,\,\sup_{w\in
E\setminus\\{0\\}}\frac{\displaystyle\int_{\Omega}\lvert\nabla
w\rvert^{2}\,dx}{\displaystyle\int_{\Omega}w^{2}\,dx},\qquad\qquad\sigma_{k}(\Omega)=\inf_{\begin{subarray}{c}E\in\mathcal{S}_{k+1}(\Omega)\end{subarray}}\,\,\sup_{w\in
E\setminus\\{0\\}}\frac{\displaystyle\int_{\Omega}\lvert\nabla
w\rvert^{2}\,dx}{\displaystyle\int_{\partial\Omega}w^{2}\,d\mathcal{H}^{n-1}},$

where $\mathcal{S}_{k+1}(\Omega)$ is the family of all linear subspaces of
$H^{1}(\Omega)$ of dimension $k+1$. In particular, we are interested in the
principal eigenvalues, i.e. $k=1$, namely

$\mu_{1}(\Omega)=\inf_{\begin{subarray}{c}w\in
H^{1}(\Omega)\setminus\\{0\\}\\\
\int_{\Omega}w=0\end{subarray}}\frac{\displaystyle\int_{\Omega}\lvert\nabla
w\rvert^{2}\,dx}{\displaystyle\int_{\Omega}w^{2}\,dx},$

$\sigma_{1}(\Omega)=\inf_{\begin{subarray}{c}w\in
H^{1}(\Omega)\setminus\\{0\\}\\\
\int_{\partial\Omega}w=0\end{subarray}}\frac{\displaystyle\int_{\Omega}\lvert\nabla
w\rvert^{2}\,dx}{\displaystyle\int_{\partial\Omega}w^{2}\,d\mathcal{H}^{n-1}}.$

Many authors in the literature identified remarkable similarities between the
two families of eigenvalues. Moreover, an underlying relationship holds
between the two quantities. For instance, Steklov eigenvalues can be seen as
limits of weighted Neumann eigenvalues, while Neumann eigenvalues can be
obtained as limits of Steklov eigenvalues by suitably perforating the set
$\Omega$. We refer, for instance, to [8], and [4] for these results. We want
to explore the same relationship between the two eigenvalues, from the shape
optimization point of view.

Namely, we could be interested in the scale invariant ratio

$F(\Omega)=\frac{\lvert\Omega\rvert\mu_{1}(\Omega)}{P(\Omega)\sigma_{1}(\Omega)},$

and, consequently, in the two problems

$\min_{\Omega\in\mathcal{K}}F(\Omega),\qquad\qquad\max_{\Omega\in\mathcal{K}}F(\Omega),$
(1.1)

where $\mathcal{K}$ is a suitable class of sets, $\lvert\cdot\rvert$ denotes
the area, and $P(\cdot)$ denotes the perimeter. Unfortunately, the choice

$\mathcal{K}=\Set{\Omega\subset\mathbb{R}^{2}}{\Omega\text{ bounded, open and
Lipschitz}}$

causes the problems in (1.1) to be ill-posed, in the sense that

$\inf_{\mathcal{K}}F(\Omega)=0,\qquad\qquad\sup_{\mathcal{K}}F(\Omega)=+\infty,$

as shown in [6], [2], and [5].

In order to obtain some comparison between Neumann and Steklov eigenvalues, we
address the problems in (1.1) restricting the class of admissible sets to

$\mathcal{K}_{c}=\Set{\Omega\subset\mathbb{R}^{2}}{\Omega\text{ bounded, open
and convex}}\\!.$ (1.2)

This choice of $\mathcal{K}_{c}$ avoids shapes that could make $F$ degenerate,
and precisely it could be shown, as in [6], that there exist two constants
$c,C>0$ such that

$c\leq F(\Omega)\leq C\qquad\forall\,\Omega\in\mathcal{K}_{c}.$

Additionally, numerical simulations lead the authors to state the following

######  Conjecture  (Henrot, Michetti [6]).

Let $\mathcal{K}_{c}$ be as in (1.2), then

$1<F(\Omega)<2\qquad\forall\,\Omega\in\mathcal{K}_{c}.$

Moreover, the inequalities are sharp in the following sense: there exists a
sequence $R_{n}$ of thinning rectangles, and a sequence $T_{n}$ of thinning
triangles such that

$\lim_{n}F(R_{n})=1,\qquad\qquad\lim_{n}F(T_{n})=2.$

The aim of this paper is to take steps towards proving the conjecture;
however, we do not provide an exhaustive solution.

The numerical simulations which support Section 1 also suggest that the
infimum and the supremum of $F(\Omega)$, in the class $\mathcal{K}_{c}$, are
asymptotically achieved by particular sequences of thinning domains. Therefore
we focus on the limits of $F(\Omega_{\varepsilon})$, where
$\Omega_{\varepsilon}$ is a family of thinning domains of the type (2.1).
Indeed, following in the footsteps of [6], for such a family, there exists a
non-negative concave function $h:[0,1]\to\mathbb{R}$ such that

$\lim_{\varepsilon\to
0}\mu_{1}(\Omega_{\varepsilon})=\mu_{1}(h)\qquad\qquad\lim_{\varepsilon\to
0}\frac{P(\Omega_{\varepsilon})\sigma_{1}(\Omega_{\varepsilon})}{\lvert\Omega_{\varepsilon}\rvert}=\sigma_{1}(h)\left(\int_{0}^{1}h(t)\,dt\right)^{-1},$

where $\mu_{1}(h)$ is the first eigenvalue of the Sturm-Liouville problem

$\begin{cases}-\dfrac{d}{dx}\left(h(x)\dfrac{dv}{dx}(x)\right)=\mu_{1}(h)h(x)v(x)\qquad&x\in(0,1),\\\\[15.0pt]
h(0)\dfrac{dv}{dx}(0)=h(1)\dfrac{dv}{dx}(1)=0,\end{cases}$ (1.3)

while $\sigma_{1}(h)$ is the first eigenvalue of the Sturm-Liouville problem

$\begin{cases}-\dfrac{d}{dx}\left(h(x)\dfrac{dv}{dx}(x)\right)=\sigma_{1}(h)v(x)\qquad&x\in(0,1),\\\\[15.0pt]
h(0)\dfrac{dv}{dx}(0)=h(1)\dfrac{dv}{dx}(1)=0.\end{cases}$ (1.4)

The function $h$, in some sense, represents the profile of the thinning sets
$\Omega_{\varepsilon}$, and, in particular, we have that $h\equiv 1$
represents the limit of a family of thinning rectangles. On the other hand,
for every $x_{0}\in(0,1)$, we let

$T_{x_{0}}(x):=\begin{cases}\dfrac{2x}{x_{0}}&x\in[0,x_{0}),\\\\[15.0pt]
\dfrac{2(1-x)}{1-x_{0}}&x\in[x_{0},1],\end{cases}$

and

$T_{0}(x)=2(1-x),\qquad\qquad T_{1}(x)=2x,$

which represents the limit of a family of thinning triangles. Consequently,
familiarizing oneself with the properties of $\mu_{1}(h)$ and $\sigma_{1}(h)$
can offer advantages when it comes to analyzing the eigenvalues
$\mu_{1}(\Omega)$ and $\sigma_{1}(\Omega)$. It is worth mentioning that the
quantities $\mu_{1}(h)$ and $\sigma_{1}(h)$ are in a way related to a weighted
Hardy constant (see [7], [10], [11], and Section 4).

Following this path, we refer to [11], [6] and [12] for the proof of the
subsequent properties: let

$\mathcal{P}=\Set{h\in L^{\infty}(0,1)\colon\,h\text{ non negative, concave
and not identically zero}},$

and

$\mathcal{P}_{1}=\Set{h\in\mathcal{P}}{\int_{0}^{1}h(t)\,dt=1},$

then for every $h\in\mathcal{P}_{1}$, we have that

$\begin{split}\pi^{2}=\mu_{1}(1)\leq\,\,&\mu_{1}(h)\leq\mu_{1}(T_{1/2})\\\\[5.0pt]
&\sigma_{1}(h)\leq\sigma_{1}(p)=12,\end{split}$

where $p$ is the arc of parabola $p(x)=6x(1-x)$.

Here we state the main results of this work

######  Theorem 1.1.

The minimum problem

$\min_{h\in\mathcal{P}_{1}}\sigma_{1}(h)$ (1.5)

admits the functions $T_{0}$ and $T_{1}$ as unique solutions.

We prove the theorem following two distinct approaches. Section 3 is devoted
to the former, while Section 4 is devoted to the latter, which relies on a
rearrangement method that, up to our knowledge, appears to be new. Finally, in
Section 5 we establish a relationship between $\mu_{1}(h)$ and
$\sigma_{1}(h)$.

######  Theorem 1.2.

There exists an invertible operator

$\mathcal{G}:\mathcal{P}\to\mathcal{P}$

such that, for every $h,k\in\mathcal{P}$, we have

$\left(\int_{0}^{1}h(t)\,dt\right)^{2}\mu(h)=\sigma\left(\mathcal{G}(h)\right),$
(1.6)

and

$\left(\int_{0}^{1}\frac{1}{\sqrt{k(t)}}\,dt\right)^{2}\sigma(k)=\mu(\mathcal{G}^{-1}(k)).$
(1.7)

It may help to solve problems obtained by studying (1.1) among thinning
domains, namely

$\min_{h\in\mathcal{P}}\frac{\displaystyle\mu_{1}(h)\int_{0}^{1}h(t)\,dt}{\displaystyle\sigma_{1}(h)},\qquad\qquad\max_{h\in\mathcal{P}}\frac{\displaystyle\mu_{1}(h)\int_{0}^{1}h(t)\,dt}{\displaystyle\sigma_{1}(h)}.$

In particular, we can fully solve the maximizing problem, and partially solve
the minimizing problem. We summarize these results in the following two
theorems.

######  Theorem 1.3.

Let $h\in\mathcal{P}_{1}$. Then

$\frac{\mu_{1}(h)}{\sigma_{1}(h)}\leq 2,$

and the equality holds if and only if $h=T_{x_{0}}$ for some $x_{0}\in[0,1]$.
If, in addition, $h(x)=h(1-x)$ for every $x\in[0,1]$, then

$\frac{\mu_{1}(h)}{\sigma_{1}(h)}\geq 1.$

##  2 Notations and tools

Here we define standard quantities for convex sets and the formal definition
of thin domain. This definition passes through the ones of support function
and minimal width (or thickness).

We refer to [6] for the proof of the lemmas in this section.

######  Definition .

Let $\Omega\subset\mathbb{R}^{N}$ be a bounded, open, and convex set. We
define the _support function_ of $\Omega$ as

$h_{\Omega}(y)=\sup_{x\in\Omega}\left(x\cdot y\right),\qquad
y\in\mathbb{R}^{n}.$

######  Definition .

Let $\Omega\subset\mathbb{R}^{N}$ be a bounded, open and convex set, and let
$y\in\mathbb{R}^{n}$. We define the _width_ of $\Omega$ in the direction $y$
as

$\omega_{\Omega}(y)=h_{\Omega}(y)+h_{\Omega}(-y)$

and we define the _minimal width_ of $\Omega$ as

$w_{\Omega}=\min\\{\omega_{\Omega}(y)\,|\;y\in\mathbb{S}^{n-1}\\}.$

Hence, if $\operatorname{diam}(\Omega)$ denotes the diameter of $\Omega$, then
we have

######  Definition .

Let $\Omega_{\delta}\subset\mathbb{R}^{n}$ be a family of non-empty, bounded,
open, and convex sets. We say that $\Omega_{\delta}$ is a family of _thinning
domains_ if

$\lim_{\delta\to
0}\,\dfrac{w_{\Omega_{\delta}}}{\operatorname{diam}(\Omega_{\delta})}=0.$

$\operatorname{diam}(\Omega_{l})$$w_{\Omega_{l}}$$\Omega_{l}$ Figure 1:
Minimal width and diameter of a convex set.

Let us now consider a particular family of thinning domains. Let
$h^{+},h^{-}\in\mathcal{P}$ such that $h^{+}+h^{-}\in\mathcal{P}_{1}$. We
consider the family of thinning domains

$\Omega_{\varepsilon}=\Set{(x,y)\in\mathbb{R}^{2}}{\begin{aligned}
\,0\leq\,&x\leq 1,\\\ -\varepsilon h_{-}(x)\leq\,&y\leq\varepsilon
h_{+}(x).\end{aligned}}$ (2.1)

For such a sequence we have that both the principal eigenvalues of the Neumann
and Steklov problems converge to the principal eigenvalues of the Sturm-
Liouville problems (1.3) and (1.4) respectively. More precisely, if we define

$\mu_{1}(h)=\inf_{\begin{subarray}{c}u\in H^{1}(0,1)\\\
\int_{0}^{1}uh\,dx=0\end{subarray}}\dfrac{\displaystyle\int_{0}^{1}(u^{\prime})^{2}h\,dx}{\displaystyle\int_{0}^{1}u^{2}h\,dx},$
(2.2)

$\sigma_{1}(h)=\inf_{\begin{subarray}{c}v\in H^{1}(0,1)\\\
\int_{0}^{1}v\,dx=0\end{subarray}}\dfrac{\displaystyle\int_{0}^{1}(v^{\prime})^{2}h\,dx}{\displaystyle\int_{0}^{1}v^{2}\,dx},$
(2.3)

we have the following lemmas

######  Lemma .

Let $\\{\Omega_{\varepsilon}\\}$ be family of thinning domains as in (2.1) and
let $h=h_{-}+h_{+}$. Then

$\displaystyle\mu_{1}(\Omega_{\varepsilon})$
$\displaystyle=\mu_{1}(h)+o(1)\text{ as }\varepsilon\to 0,$
$\displaystyle\sigma_{1}(\Omega_{\varepsilon})$
$\displaystyle=\frac{\sigma_{1}(h)}{2}\varepsilon+o(\varepsilon)\text{ as
}\varepsilon\to 0.$

The following compactness result for $\mathcal{P}$ holds true

######  Lemma .

Let $h_{n}\in\mathcal{P}_{1}$ be a sequence of functions, then there exists
$h\in\mathcal{P}$ such that, up to a subsequence, we have:

  * •

$h_{n}$ converges to $h$ in $L^{2}(0,1)$;

  * •

$h_{n}$ converges to $h$ uniformly on every compact subset of $(0,1)$.

We also recall a continuity property of the eigenvalues $\mu_{1}(h)$ and
$\sigma_{1}(h)$.

######  Lemma .

Let $h_{n},h\in\mathcal{P}$ be a sequence such that $h_{n}$ converges in
$L^{2}(0,1)$ to $h$. Then we have

$\displaystyle\lim_{n}\mu_{1}(h_{n})=\mu_{1}(h),$
$\displaystyle\lim_{n}\sigma_{1}(h_{n})=\sigma_{1}(h).$

###  2.1 Other tools

Here we recall some other tools that will be useful in the next pages. We
refer to [9, 1].

######  Theorem 2.1 (Coarea formula).

Let $\Omega\subset\mathbb{R}^{n}$ be an open set with Lipschitz boundary. Let
$f\in W^{1,1}_{\text{loc}}(\Omega)$, and let $u:\Omega\to\mathbb{R}$ be a
measurable function. Then,

${\displaystyle\int_{\Omega}u(x)|\nabla
f(x)|dx=\int_{\mathbb{R}}dt\int_{\Omega\cap
f^{-1}(t)}u(y)\,d\mathcal{H}^{n-1}(y)}.$

Here we define the notion of decreasing rearrangement

######  Definition .

Let $\Omega\subset\mathbb{R}^{n}$ be an open set, and let
$u:\Omega\to\mathbb{R}$ be a measurable function. We define the _distribution
function_ $\eta_{u}:[0,+\infty[\,\to[0,+\infty[$ of $u$ as the function

$\eta_{u}(t)=\lvert\Set{x\in\Omega\,:\,\lvert u(x)\rvert>t}\rvert$

######  Definition .

Let $u:\Omega\to\mathbb{R}$ be a measurable function. We define the
_increasing rearrangement_ $u_{*}$ of $u$ as

$u_{*}(s)=\inf\Set{t>0}{\eta(t)\leq\lvert\Omega\rvert-s}.$

######  Remark .

Let $\Omega\subset\mathbb{R}^{n}$ be an open set, and let
$u:\Omega\to\mathbb{R}$ be a measurable function. Then $u$ and its increasing
rearrangement $u_{*}$ are equi-measurable namely

$\eta_{u}=\eta_{u_{*}},$

and, in addition, for every $p\in[1,+\infty)$,

$\displaystyle{\lVert u\rVert_{L^{p}(\Omega)}=\lVert
u_{*}\rVert_{L^{p}(0,\lvert\Omega\rvert)}}.$

Finally, here is an important property of extreme points convex sets.

######  Definition .

Let $V$ be a vector space, let $C\subset V$ be a convex set, and let $z\in C$.
We say that $z$ is an _extreme point_ of $C$ if it cannot be written as a
convex combination of distinct elements of $C$. More precisely, if
$z=(1-t)x+ty$, with $x,y\in C$ and $t\in[0,1]$, then $x=y=z$.

######  Proposition .

Let $h\in\mathcal{P}_{1}$. Then $h$ is an extreme point for $\mathcal{P}_{1}$
if and only if there exists $x_{0}\in[0,1]$ such that $h=T_{x_{0}}$.

###### Proof.

Let us start by proving that for every $x_{0}\in[0,1]$ the triangle
$T_{x_{0}}$ is an extreme point of $\mathcal{P}_{1}$. Let
$h\in\mathcal{P}_{1}$ and let $x_{M}$ be a maximum point for $h$, then the
concavity of $h$ ensures

$h\geq\dfrac{h(x_{M})}{2}T_{x_{M}}.$

Recalling that $\int_{0}^{1}h\,dx=1$, we get that

$h(x_{M})=\max_{[0,1]}h\leq 2,$ (2.4)

and the equality holds if and only if $h=T_{x_{M}}$.

Let now $x_{0}\in[0,1]$, and assume that

$T_{x_{0}}(x)=(1-t)h_{0}(x)+th_{1}(x)\qquad x\in[0,1],$

with $h_{0},h_{1}\in\mathcal{P}_{1}$, and $t\in[0,1]$. Since

$2=T_{x_{0}}(x_{0})\leq\max\\{h_{0}(x_{0}),h_{1}(x_{0})\\},$

and

$2=(1-t)h_{0}(x_{0})+th_{1}(x_{0}),$

we get equality in (2.4) for both $h_{0}$ and $h_{1}$. Therefore,
$h_{0}=h_{1}=T_{x_{0}}$, and we have proved that $T_{x_{0}}$ is an extreme
point of $\mathcal{P}_{1}$.

We now prove that the triangles are the only extreme points of
$\mathcal{P}_{1}$. Let $h\in\mathcal{P}_{1}$ be such that $h\neq T_{x_{0}}$
for every $x_{0}\in[0,1]$.

We begin by assuming that $h(1)>0$. Notice that, in this setting, there exists
$s\in(0,1)$, such that the function

$h_{s}=\frac{h-sT_{0}}{1-s}\in\mathcal{P}_{1}.$

In particular, we get

$h=(1-s)h_{s}+sT_{0},$

that is, $h$ is not an extreme point of $\mathcal{P}$. An analogous
computation can be done when $h(0)>0$.

Assume now that $h(0)=h(1)=0$ and let $\nu$ be the positive Radon measure
representing $-h^{\prime\prime}$. Since $h\neq T_{x_{0}}$ for every $x_{0}$,
then there exists $y_{0}\in(0,1)$ such that $\nu([0,y_{0}])>0$ and
$\nu((y_{0},1])>0$. Let

$\displaystyle\nu_{1}=\nu|_{[0,x_{0}]},$
$\displaystyle\nu_{2}=\nu|_{(x_{0},1]},$

and let $h_{1},h_{2}$ be the solutions to

$\displaystyle\begin{cases}-h_{1}^{\prime\prime}=\nu_{1},\\\
h_{1}(0)=h_{1}(1)=0,\end{cases}$
$\displaystyle\begin{cases}-h_{2}^{\prime\prime}=\nu_{2},\\\
h_{2}(0)=h_{2}(1)=0.\end{cases}$

We have that $h_{1},h_{2}\in\mathcal{P}$ and $h=h_{1}+h_{2}$, so that, letting

$\tilde{h}_{i}=\dfrac{h_{i}}{\int_{0}^{1}h_{i}\,dx},\qquad i=1,2,$

we get $\tilde{h}_{1},\tilde{h_{2}}\in\mathcal{P}_{1}$, and

$h=t\tilde{h}_{1}+(1-t)\tilde{h}_{2},$

with $t\in(0,1)$. Hence, $h$ is not an extreme point of $\mathcal{P}_{1}$. ∎

Finally, we recall the definition of a quasiconvex function.

######  Definition .

A function $f:\mathbb{R}\to\mathbb{R}$ is quasiconcave if for all
$x,y\in\mathbb{R}$ and $\lambda\in[0,1]$ we have

$f(\lambda x+(1-\lambda)y)\geq\min\big{\\{}f(x),f(y)\big{\\}}.$

A function defined on an interval is quasiconcave if and only if it is
monotone or ’increasing then decreasing’, i.e. if there are two complementary
intervals (one of which may be empty) such that it is increasing on the former
and decreasing on the latter.

##  3 Minimization of the Steklov eigenvalue

For every $h\in\mathcal{P}_{1}$ we consider the Sturm-Liouville eigenvalue
$\sigma_{1}(h)$ defined in (2.3)

Lemmas Section 2 and Section 2 prove that the problems

$\begin{split}\max&\Set{\sigma_{1}(h)\colon h\in\mathcal{P}_{1}}\\\\[15.0pt]
\min&\Set{\sigma_{1}(h)\colon h\in\mathcal{P}_{1}}\end{split}$

admit solutions. In particular, the solution to the maximization problem (see
for instance [12]) is given by the parabola $p(x)=6x(1-x)$, with corresponding
eigenvalue $\sigma_{1}(p)=12$. In this section, we aim to prove Theorem 1.1,
namely that the problem

$\min\Set{\sigma_{1}(h)\colon h\in\mathcal{P}_{1}},$

admits as unique solutions the functions $T_{0}(x)=2(1-x)$ and $T_{1}(x)=2x$
with corresponding eigenvalue

$\sigma_{1}(T_{0})=\sigma_{1}(T_{1})=(j^{\prime}_{0,1})^{2}/2,$

where $j^{\prime}_{0,1}$ is the first positive zero of the first derivative of
the Bessel function $J_{0}$.

######  Remark .

The function

$h\in\mathcal{P}\longmapsto\sigma_{1}(h),$

satisfies the following properties:

  * •

monotonicity: for every $h_{0},h_{1}\in\mathcal{P}$, if $h_{0}\leq h_{1}$ then

$\sigma_{1}(h_{0})\leq\sigma_{1}(h_{1});$

  * •

homogeneity: for every $h\in\mathcal{P}$ and for every $\alpha>0$,

$\sigma_{1}(\alpha h)=\alpha\sigma_{1}(h);$

  * •

concavity: for every $h_{0},h_{1}\in\mathcal{P}$ and for every $t\in[0,1]$,
letting $h_{t}=(1-t)h_{0}+th_{1}$, we have that

$\sigma_{1}(h_{t})\geq(1-t)\sigma_{1}(h_{0})+t\,\sigma_{1}(h_{1});$

  * •

symmetry: let $h\in\mathcal{P}$, and let $k(x)=h(1-x)$, then

$\sigma_{1}(k)=\sigma_{1}(h).$ (3.1)

######  Proposition .

Let $h\in\mathcal{P}_{1}$ be a solution to problem (1.5), then $h$ is an
extreme point of $\mathcal{P}_{1}$.

###### Proof.

Let $h\in\mathcal{P}_{1}$ be a solution to problem (1.5). By contradiction
assume that $h$ is not an extreme point of $\mathcal{P}_{1}$. Let
$h_{0},h_{1}\in\mathcal{P}_{1}\setminus\set{h}$ and $t\in(0,1)$ such that

$h=(1-t)h_{0}+th_{1}.$

Let $v\in H^{1}(0,1)$ be an eigenfunction for $\sigma_{1}(h)$ with

$\int_{0}^{1}v^{2}\,dx=1,$

then

$\begin{split}\sigma_{1}(h)&=\int_{0}^{1}(v^{\prime})^{2}h\,dx=(1-t)\int_{0}^{1}(v^{\prime})^{2}h_{0}\,dx+t\int_{0}^{1}(v^{\prime})^{2}h_{1}\,dx\\\\[10.0pt]
&\geq(1-t)\sigma_{1}(h_{0})+t\sigma_{1}(h_{1}).\end{split}$

On the other hand, by the minimality of $\sigma_{1}(\Omega)$, we have

$\displaystyle\sigma_{1}(h_{0})=\int_{0}^{1}(v^{\prime})^{2}h_{0}\,dx,$
$\displaystyle\sigma_{1}(h_{1})=\int_{0}^{1}(v^{\prime})^{2}h_{1}\,dx.$

Therefore, $v$ is also an eigenfunction for $\sigma_{1}(h_{0})$ and
$\sigma_{1}(h_{1})$. Let us now prove that $h_{0}=h$, thus reaching a
contradiction. From the weak form of equation (1.4), we have that for every
$\varphi\in H^{1}(0,1)$

$\begin{split}\int_{0}^{1}v^{\prime}\varphi^{\prime}h\,dx&=\sigma_{1}(h)\int_{0}^{1}v\varphi\,dx\\\\[10.0pt]
&=\sigma_{1}(h_{0})\int_{0}^{1}v\varphi\,dx=\int_{0}^{1}v^{\prime}\varphi^{\prime}h_{0}\,dx,\end{split}$

that is

$\int_{0}^{1}(h-h_{0})v^{\prime}\varphi^{\prime}\,dx=0$

for every $\varphi\in H^{1}(0,1)$, which yields $h=h_{0}$, since, for every
$\psi\in L^{2}(0,1)$, we can choose

$\varphi(x)=\int_{0}^{x}\psi(t)\,dt.$

∎

In order to study the minimum problem (1.5), we need to evaluate $\sigma_{1}$
on triangles, and we will need the following result, whose proof can be found
in [6].

######  Lemma .

Let $x_{0}\in[0,1]$. Then $\sigma_{1}(T_{x_{0}})$ is the first non-zero root
$\sigma$ of the equation

$J_{0}\left(\sqrt{2\sigma}x_{0}\right)J_{0}^{\prime}\left(\sqrt{2\sigma}(1-x_{0})\right)+J_{0}\left(\sqrt{2\sigma}(1-x_{0})\right)J_{0}^{\prime}\left(\sqrt{2\sigma}x_{0}\right)=0.$
(3.2)

In addition, here we summarize the properties of the Bessel function $J_{0}$
which we will use.

######  Proposition .

Let $J_{0}$ be the Bessel function of the first kind of order 0, and let
$j_{0,1}$ and $j_{0,1}^{\prime}$ be the first zero of $J_{0}$ and
$J_{0}^{\prime}$ respectively. Then

$0<j_{0,1}<j_{0,1}^{\prime},$

and

$\displaystyle J_{0}(x)\geq 0\qquad$ $\displaystyle\forall\,x\in(0,j_{0,1}),$
$\displaystyle J_{0}^{\prime}(x)\leq 0$
$\displaystyle\forall\,x\in(0,j_{0,1}^{\prime}),$ $\displaystyle J_{0}(x)<0$
$\displaystyle\forall\,x\in(j_{0,1},j_{0,1}^{\prime}).$

###### Proof of Theorem 1.1.

By Section 2 and Section 2 we have that the minimum problem (1.5) admits a
solution. On the other hand, by Section 3 and Subsection 2.1 we have that a
solution to (1.5) has to be a triangle $T_{x_{0}}$ for some $x_{0}\in[0,1]$.
By the symmetry of $\sigma_{1}$ stated in (3.1), we notice that to prove the
theorem it is sufficient to show that the function

$x_{0}\in\left[0,\frac{1}{2}\right]\mapsto\sigma_{1}(T_{x_{0}}),$

attains its minimum for $x_{0}=0$.

Let $j_{0,1}$ and $j^{\prime}_{0,1}$ be the first positive roots of $J_{0}$
and $J^{\prime}_{0}$ respectively. For every $x\in[0,1/2]$, and
$s\in[0,+\infty)$, let

$F(x,s)=J_{0}(sx)J_{0}^{\prime}(s(1-x))+J_{0}(s(1-x))J_{0}^{\prime}(sx),$

which is the function defined in Section 3 that determines the value
$\sigma_{1}(T_{x_{0}})$. Let $x_{0}\in(0,1/2)$ and let $s(x_{0})$ be the
smallest non-zero root of the equation

$F(x_{0},s)=0.$ (3.3)

We claim that

$s(x_{0})\in
I_{x_{0}}=\left(\dfrac{j_{0,1}}{(1-x_{0})},\min\Set{\dfrac{j_{0,1}}{x_{0}},\dfrac{j^{\prime}_{0,1}}{1-x_{0}}}\right).$
(3.4)

Indeed, since $J_{0}$ and $-J_{0}^{\prime}$ are positive in $(0,j_{0,1})$, and
$x_{0}<1-x_{0}$, then

$F(x_{0},s)<0\qquad\forall\,s\in\left(0,\frac{j_{0,1}}{1-x_{0}}\right].$

On the other hand, using again the properties in Section 3, a direct
computation gives

$F\left(x_{0},\min\Set{\dfrac{j_{0,1}}{x_{0}},\dfrac{j^{\prime}_{0,1}}{1-x_{0}}}\right)>0,$

thus proving the claim. Notice that (3.4) gives

$\displaystyle J_{0}(s(x_{0})x_{0})>0,\qquad$ $\displaystyle
J_{0}(s(x_{0})(1-x_{0}))<0,$ (3.5) $\displaystyle
J_{0}^{\prime}(s(x_{0})x_{0})<0,\qquad$ $\displaystyle
J_{0}^{\prime}(s(x_{0})(1-x_{0}))<0.$

Since $J_{0}$ solves the equation

$J_{0}^{\prime\prime}(t)+\frac{J_{0}^{\prime}(t)}{t}+J_{0}(t)=0,$ (3.6)

then we have

$\begin{split}\partial_{s}F(x_{0},s)=&J^{\prime}_{0}(sx_{0})J_{0}^{\prime}(s(1-x_{0}))-J_{0}(sx_{0})J_{0}(s(1-x_{0}))\\\\[8.0pt]
&-\dfrac{1}{s}\left(J_{0}(sx_{0})J^{\prime}_{0}(s(1-x_{0}))+J_{0}(s(1-x_{0}))J^{\prime}_{0}(sx_{0})\right).\end{split}$

In particular, (3.3) and (3.5) ensure that

$\partial_{s}F(x_{0},s(x_{0}))>0.$ (3.7)

By the implicit function theorem, the function $x_{0}\mapsto s(x_{0})$ is
continuous, differentiable and

$s^{\prime}(x_{0})\partial_{s}F(x_{0},s(x_{0}))+\partial_{x}F(x_{0},s(x_{0}))=0.$

Using (3.6), direct computations give

$\partial_{x}F(x_{0},s(x_{0}))=-\dfrac{J_{0}(s(x_{0})(1-x_{0}))J^{\prime}_{0}(s(x_{0})x_{0})}{x_{0}}+\dfrac{J(s(x_{0})x_{0})J^{\prime}_{0}(s(x_{0})(1-x_{0}))}{1-x_{0}}.$

As before, (3.5) ensure that

$\partial_{x}F(x_{0},s(x_{0}))<0.$ (3.8)

Joining (3.7) and (3.8), we have that $s^{\prime}(x_{0})>0$ and $x_{0}\mapsto
s(x_{0})$ is increasing. Finally,

$\sigma_{1}(T_{x_{0}})=s^{2}(x_{0})/2$

is increasing for $x_{0}\in(0,1/2)$, and the minimum is achieved when
$x_{0}=0$. ∎

######  Remark .

Equation (3.2) for $x_{0}=0$ reduces to

$J^{\prime}_{0}\left(\sqrt{2\sigma}\right)=0$

that is, $\sigma_{1}(2x)=\sigma_{1}(T_{0})=(j^{\prime}_{0,1})^{2}/2$.

##  4 An alternative proof for the minimum of $\sigma(h)$

In this section, we minimize $\sigma_{1}(h)$ using an alternative approach
that avoids the explicit computation of the eigenvalue. In particular, our aim
is to define a particular kind of symmetrization that allows us to prove that
solutions to (1.5) have to be monotone. Before defining the aforementioned
symmetrization we prove an equivalent formulation for the eigenvalue
$\sigma_{1}(h)$, referring to the ideas for the proof in [10, Lemma 4.2]

######  Proposition .

Let $h\in\mathcal{P}_{1}$, then

$\sigma_{1}(h)=\min\Set{\dfrac{\displaystyle\int_{0}^{1}(\varphi^{\prime})^{2}\,dx}{\displaystyle\int_{0}^{1}\frac{\varphi^{2}}{h}\,dx}}{\begin{aligned}
&\>\,\varphi\in H^{1}_{0}(0,1),\\\\[3.0pt]
\displaystyle&\int_{0}^{1}\frac{\varphi^{2}}{h}\,dx<\infty\end{aligned}}.$

###### Proof.

Let $v\in H^{1}(0,1)$ be a weak solution to (1.4) and let $w=hv^{\prime}$.
Then, since $w^{\prime}=-\sigma_{1}(h)v$, we have that $w\in H^{2}(0,1)\cap
H^{1}_{0}(0,1)$ and that is a solution to

$\begin{cases}\displaystyle{-w^{\prime\prime}(x)=\frac{\sigma_{1}(h)}{h(x)}w(x)}\qquad\qquad
x\in(0,1)\\\\[7.0pt] w(0)=w(1)=0,\end{cases}$ (4.1)

and

$\dfrac{\displaystyle\int_{0}^{1}(w^{\prime})^{2}\,dx}{\displaystyle\int_{0}^{1}\frac{w^{2}}{h}\,dx}=\sigma_{1}(h).$
(4.2)

Following classical arguments (see for instance [3]) we have that $v$ vanishes
in one and only one point $x_{0}\in(0,1)$, so that $w^{\prime}$ vanishes only
in $x_{0}$. Without loss of generality, we can assume that $w^{\prime}$ is
positive in $[0,x_{0})$ and it is negative in $(x_{0},1]$. Let now $\varphi\in
H^{1}_{0}(0,1)$ be such that

$\int_{0}^{1}\dfrac{\varphi^{2}}{h}\,dx<+\infty.$

Then, for every $0<x<x_{0}$, we have that

$\begin{split}\dfrac{1}{h(x)}\varphi^{2}(x)&\leq\dfrac{1}{h(x)}\left(\int_{0}^{x}\dfrac{(\varphi^{\prime}(t))^{2}}{w^{\prime}(t)}\,dt\right)\left(\int_{0}^{x}w^{\prime}(t)\,dt\right)\\\\[15.0pt]
&=\dfrac{w(x)}{h(x)}\int_{0}^{x}\dfrac{(\varphi^{\prime}(t))^{2}}{w^{\prime}(t)}\,dt.\end{split}$

Then, integrating from $0$ to $x_{1}<x_{0}$, we get

$\begin{split}\int_{0}^{x_{1}}\dfrac{\varphi^{2}(x)}{h(x)}\,dx&\leq\int_{0}^{x_{1}}\dfrac{w(x)}{h(x)}\int_{0}^{x}\dfrac{(\varphi^{\prime}(t))^{2}}{w^{\prime}(t)}\,dt\,dx\\\\[15.0pt]
&=\int_{0}^{x_{1}}\dfrac{(\varphi^{\prime}(t))^{2}}{w^{\prime}(t)}\int_{t}^{x_{1}}\dfrac{w(x)}{h(x)}\,dx\,dt.\end{split}$

Using (4.1), and the fact that $w^{\prime}(x_{1})>0$, then we have

$\int_{0}^{x_{1}}\dfrac{\varphi^{2}(x)}{h(x)}\,dx\leq\dfrac{1}{\sigma_{1}(h)}\int_{0}^{x_{1}}(\varphi^{\prime}(t))^{2}\,dt.$

Letting $x_{1}$ go to $x_{0}$ we have

$\int_{0}^{x_{0}}\dfrac{\varphi^{2}(x)}{h(x)}\,dx\leq\dfrac{1}{\sigma_{1}(h)}\int_{0}^{x_{0}}(\varphi^{\prime}(x))^{2}\,dx.$
(4.3)

Similar computations can be done in the case $x>x_{0}$, so that we have

$\int_{x_{0}}^{1}\dfrac{\varphi^{2}(x)}{h(x)}\,dx\leq\dfrac{1}{\sigma_{1}(h)}\int_{x_{0}}^{1}(\varphi^{\prime}(x))^{2}\,dx.$
(4.4)

Joining (4.3) and (4.4) we have

$\int_{0}^{1}\dfrac{\varphi^{2}(x)}{h(x)}\,dx\leq\dfrac{1}{\sigma_{1}(h)}\int_{0}^{1}(\varphi^{\prime}(x))^{2}\,dx,$

that is

$\dfrac{\displaystyle\int_{0}^{1}(\varphi^{\prime})^{2}\,dx}{\displaystyle\int_{0}^{1}\frac{\varphi^{2}}{h}\,dx}\geq\sigma_{1}(h).$
(4.5)

Since $w$ is an admissible function, the assertion follows from (4.2) and
(4.5). ∎

We now define the rearrangement mentioned above. Let

$w:[0,1]\to\mathbb{R}$

be a quasi-concave piecewise $C^{1}$ function such that

$\lvert\\{w^{\prime}=0\\}\rvert=0,s\qquad w(0)=w(1)=0,$

and let us denote by

$w_{M}=\max_{[0,1]}w,$

and by $x_{M}$ the maximum point of $w$. We aim to rearrange $w$ in such a way
that the derivative of the rearranged function $w^{\sharp}$ concentrates at
the left of the new maximum point $x_{M}^{*}$.

For every $t\in(0,w_{M})$, we define $(x_{t},y_{t}):=\left\\{w(x)>t\right\\}$,
and the distribution functions

$\begin{split}\eta_{1}(t)=x_{M}-x_{t}=\lvert\set{w>t}\cap(0,x_{M})\rvert,\\\
\eta_{2}(t)=y_{t}-x_{M}=\lvert\set{w>t}\cap(x_{M},1)\rvert.\end{split}$ (4.6)

Notice that

$\eta_{1}:(0,w_{M})\to(0,x_{M}),\qquad\qquad\eta_{2}:(0,w_{M})\to(0,1-x_{M})$

are both decreasing, invertible, absolutely continuous functions, and that,
for a.e. $t\in[0,1]$,

$\displaystyle\eta_{1}^{\prime}(t)=-\frac{1}{\lvert
w^{\prime}(x_{t})\rvert},\qquad\qquad\eta_{2}^{\prime}(t)=-\frac{1}{\lvert
w^{\prime}(y_{t})\rvert}.$

Let us now define the rearranged distribution functions in such a way that,
for a.e. $t\in[0,1]$,

$\begin{gathered}\eta_{\ast,1}^{\prime}(t)=\max\\{\eta_{1}^{\prime}(t),\eta_{2}^{\prime}(t)\\},\\\
\eta_{\ast,2}^{\prime}(t)=\min\\{\eta_{1}^{\prime}(t),\eta_{2}^{\prime}(t)\\},\end{gathered}$
(4.7)

namely,

$\begin{split}\eta_{\ast,1}(t):=-\int_{t}^{w_{M}}\max\\{\eta_{1}^{\prime}(s),\eta_{2}^{\prime}(s)\\}\,ds,\\\
\eta_{\ast,2}(t):=-\int_{t}^{w_{M}}\min\\{\eta_{1}^{\prime}(s),\eta_{2}^{\prime}(s)\\}\,ds.\end{split}$
(4.8)

######  Remark .

Here we emphasize some properties of these distribution functions:

  * •

for every $t\in(0,w_{M})$, we have

$\eta_{1}(t)+\eta_{2}(t)=\eta_{\ast,1}(t)+\eta_{\ast,2}(t)=\lvert\\{w>t\\}\rvert;$

  * •

by (4.7), we have that, for a.e. $t\in(0,w_{M})$,

$\begin{split}\frac{1}{\lvert\eta_{\ast,1}^{\prime}(t)\rvert}&=\max\left\\{\frac{1}{\lvert\eta_{1}^{\prime}(t)\rvert},\frac{1}{\lvert\eta_{2}^{\prime}(t)\rvert}\right\\}\\\\[5.0pt]
&=\max\\{\lvert w^{\prime}(x_{t})\rvert,\lvert
w^{\prime}(y_{t})\rvert\\},\end{split}$

and

$\begin{split}\frac{1}{\lvert\eta_{\ast,2}^{\prime}(t)\rvert}&=\min\left\\{\frac{1}{\lvert\eta_{1}^{\prime}(t)\rvert},\frac{1}{\lvert\eta_{2}^{\prime}(t)\rvert}\right\\}\\\\[5.0pt]
&=\min\\{\lvert w^{\prime}(x_{t})\rvert,\lvert
w^{\prime}(y_{t})\rvert\\}.\end{split}$

  * •

By (4.7), we have

$\frac{1}{\lvert\eta_{\ast,1}^{\prime}(t)\rvert^{\alpha}}+\frac{1}{\lvert\eta_{\ast,2}^{\prime}(t)\rvert^{\alpha}}=\frac{1}{\lvert\eta_{1}^{\prime}(t)\rvert^{\alpha}}+\frac{1}{\lvert\eta_{2}^{\prime}(t)\rvert^{\alpha}}$
(4.9)

for every $\alpha\in\mathbb{R}$.

  * •

for $t=0$, we denote by

$x^{\ast}_{M}:=\eta_{\ast,1}(0)=1-\eta_{\ast,2}(0),$

this will play the role of the maximum point for the rearranged function.

  * •

the functions

$\eta_{\ast,1}:(0,w_{M})\to(0,x^{\ast}_{M}),\qquad\qquad\eta_{\ast,2}:(0,w_{M})\to(0,1-x^{\ast}_{M})$

are decreasing, invertible, absolutely continuous functions.

We now define the rearrangement $w^{\sharp}$ as follows:

######  Definition .

Let $w$ be a quasi-concave piecewise $C^{1}$ function such that

$\lvert\\{w^{\prime}=0\\}\rvert=0,\qquad w(0)=w(1)=0,$

and let $\eta_{1},\eta_{2},\eta_{\ast,1}$ and $\eta_{\ast,2}$ be the functions
defined in (4.6), and (4.8). We define the competitor $w^{\\#}$ as

$w^{\sharp}(x)=\begin{cases}w_{m}-\eta_{\ast,1}^{-1}(x)\quad&\text{ if }x\leq
x^{\ast}_{M},\\\ w_{m}-\eta_{\ast,2}^{-1}(1-x)\quad&\text{ if
}x>x^{\ast}_{M}.\end{cases}$

Figure 2: Function $w^{\\#}$ when $w$ is a quasi-concave affine function

######  Remark .

From the definition we have that $w^{\sharp}$ is increasing in
$[0,x_{M}^{\ast})$ and decreasing in $(x_{M}^{\ast},0]$, so that $w^{\sharp}$
is quasi-concave. Moreover, we have that $w^{\sharp}$ and $w$ are equi-
measurable, i.e.

$\lVert w^{\sharp}\rVert_{L^{p}(0,1)}=\lVert w\rVert_{L^{p}(0,1)}$

for every $p\in[1,+\infty]$.

We now prove some useful properties of this rearrangement.

######  Lemma .

Let $w$ be a quasi-concave piecewise $C^{1}$ function such that

$\lvert\\{w^{\prime}=0\\}\rvert=0,\qquad w(0)=w(1)=0,$

and let $w^{\sharp}$ be the competitor defined in Section 4. Then

$w^{\sharp}(x)=(w(1-x))^{\sharp}.$

###### Proof.

Let us set $v(x)=w(1-x)$ and $\nu_{1},\nu_{2},\nu_{\ast,1}$ and $\nu_{\ast,2}$
the equivalent quantities defined for $v$. Then we have

$\nu^{\prime}_{1}(t)=\eta^{\prime}_{2}(t),\qquad\qquad\nu^{\prime}_{2}(t)=\eta^{\prime}_{1}(t),$

and, in particular,

$\nu^{\prime}_{\ast,1}(t)=\eta^{\prime}_{\ast,1}(t),\qquad\qquad\nu^{\prime}_{\ast,2}(t)=\eta^{\prime}_{\ast,2}(t).$

∎

######  Lemma .

Let $w$ be a quasi-concave piecewise $C^{1}$ function such that

$\lvert\\{w^{\prime}=0\\}\rvert=0,\qquad w(0)=w(1)=0,$

and let $w^{\sharp}$ be its competitor defined in Section 4. Then,

$\lVert(w^{\sharp})^{\prime}\rVert_{L^{p}(0,1)}=\lVert
w^{\prime}\rVert_{L^{p}(0,1)}\qquad\forall\,p\geq 1.$ (4.10)

###### Proof.

Let us compute separately the norms: by the coarea formula (see LABEL:coarea),
we get

$\begin{split}\int_{0}^{1}\lvert
w^{\prime}(x)\rvert^{p}\,dx&=\int_{0}^{w_{M}}\int_{\\{w=t\\}}\lvert
w^{\prime}(x)\rvert^{p-1}\,d\mathcal{H}^{0}(x)\,dt\\\\[7.0pt]
&=\int_{0}^{w_{M}}\left(\lvert w^{\prime}(x_{t})\rvert^{p-1}+\lvert
w^{\prime}(y_{t})\rvert^{p-1}\right)\,dt\\\\[7.0pt]
&=\int_{0}^{w_{M}}\left(\frac{1}{\lvert\eta_{1}^{\prime}(t)\rvert^{p-1}}+\frac{1}{\lvert\eta_{2}^{\prime}(t)\rvert^{p-1}}\right)\,dt.\end{split}$
(4.11)

Analogously,

$\int_{0}^{1}\lvert(w^{\sharp})^{\prime}(x)\rvert^{p}\,dx=\int_{0}^{w_{M}}\frac{1}{\lvert\eta_{\ast,1}^{\prime}(t)\rvert^{p-1}}+\frac{1}{\lvert\eta_{\ast,2}^{\prime}(t)\rvert^{p-1}}\,dt.$
(4.12)

Joining (4.11), (4.12), and (4.9), we get (4.10). ∎

We now state the property of $w^{\sharp}$ that will be crucial in the proof of
Theorem 1.1.

######  Lemma .

Let $w$ be a quasi-concave piecewise $C^{1}$ function such that

$\lvert\\{w^{\prime}=0\\}\rvert=0,\qquad w(0)=w(1)=0,$

and let $w^{\sharp}$ be its competitor defined in Section 4. Assume that

$h:(0,1)\to[0,+\infty)$

is a concave function, and let $h^{\ast}$ be its increasing rearrangement.
Then

$\int_{0}^{1}\frac{w^{2}}{h}\,dx\leq\int_{0}^{1}\frac{(w^{\sharp})^{2}}{h^{\ast}}\,dx.$

###### Proof.

By Fubini’s theorem, we can write

$\int_{0}^{1}\frac{w^{2}(x)}{h(x)}\,dx=\int_{0}^{1}w^{2}(x)\int_{0}^{\frac{1}{h(x)}}\,dt\,dx=\int_{0}^{\infty}\int_{\left\\{\frac{1}{h(x)}>t\right\\}}w^{2}(x)\,dx\,dt.$

The same computation leads to

$\int_{0}^{1}\frac{(w^{\sharp})^{2}(x)}{h^{\ast}(x)}\,dx=\int_{0}^{\infty}\int_{\left\\{\frac{1}{h^{\ast}(x)}>t\right\\}}(w^{\sharp})^{2}(x)\,dx\,dt.$

Hence, to prove the lemma it is sufficient to prove that for a.e. $t>0$

$\int_{\left\\{\frac{1}{h(x)}>t\right\\}}w^{2}(x)\,dx\leq\int_{\left\\{\frac{1}{h^{\ast}(x)}>t\right\\}}(w^{\sharp})^{2}(x)\,dx.$

For every $t\in(0,\lVert 1/h\rVert_{\infty})$, let us define

$D_{t}:=\left\\{\frac{1}{h(x)}>t\right\\}=(0,\tilde{x}_{t})\cup(\tilde{y}_{t},1),$

for some $\tilde{x}_{t},\tilde{y}_{t}\in(0,1)$. In an analogous way, by the
definition of increasing rearrangement (see Subsection 2.1), we have

$D^{\ast}_{t}=\left\\{\frac{1}{h^{\ast}(x)}>t\right\\}=(0,\tilde{x}_{t}+1-\tilde{y}_{t}).$

Let $m=\min\\{w(\tilde{x}_{t}),w(\tilde{y}_{t})\\}$, and let us define the
following auxiliary functions

$f(x)=\min\\{w(x),m\\}^{2},\qquad\qquad g(x)=(w^{2}-m^{2})_{+},$

so that

$\int_{D_{t}}w^{2}\,dx=\int_{D_{t}}f\,dx+\int_{D_{t}}g\,dx.$

Similarly, we define

$f_{0}(x)=\min\\{w^{\sharp}(x),m\\}^{2},\qquad\qquad
g_{0}(x)=((w^{\sharp})^{2}-m^{2})_{+},$

so that

$\int_{D_{t}^{*}}(w^{\sharp})^{2}\,dx=\int_{D_{t}^{*}}f_{0}\,dx+\int_{D_{t}^{*}}g_{0}\,dx.$

We now evaluate separately the two terms:

  1. 1.

By the definition of $m$, we have that

$w(x)>m\qquad\forall\,x\in(0,1)\setminus D_{t}.$

Therefore, since $f$ and $f_{0}$ are equi-measurable, we get

$\begin{split}\int_{D_{t}}f(x)\,dx&=\int_{0}^{1}f(x)\,dx-(1-\lvert
D_{t}\rvert)m^{2}\\\\[7.0pt] &=\int_{0}^{1}f_{0}(x)\,dx-\int_{(0,1)\setminus
D^{\ast}_{t}}m^{2}\,dx\\\\[7.0pt]
&\leq\int_{0}^{1}f_{0}(x)\,dx-\int_{(0,1)\setminus
D^{\ast}_{t}}f_{0}(x)\,dx\\\\[7.0pt]
&=\int_{D^{\ast}_{t}}f_{0}(x)\,dx,\end{split}$ (4.13)

where we have used that $\lvert D_{t}\rvert=\lvert D_{t}^{*}\rvert$, and that
$m\geq f_{0}$;

  2. 2.

Section 4 allows us to assume without loss of generality that
$w(\tilde{y}_{t})=m$. Therefore, the quasi-concavity of $w$ ensures that

$w(x)\leq m\qquad\forall\,x\in(\tilde{y}_{t},1),$

and we can write

$\int_{D_{t}}g(x)\,dx=\int_{0}^{\tilde{x}_{t}}(w^{2}(x)-m^{2})_{+}\,dx=\int_{m}^{w_{M}}2r\lvert\\{w>r\\}\cap(0,\tilde{x}_{t})\rvert\,dr.$
(4.14)

On the other hand,

$\begin{split}\int_{D^{\ast}_{t}}g_{0}(x)\,dx&=\int_{0}^{\tilde{x}_{t}+1-\tilde{y}_{t}}g_{0}(x)\,dx\\\\[7.0pt]
&\geq\int_{0}^{\tilde{x}_{t}}g_{0}(x)\,dx\\\\[7.0pt]
&=\int_{m}^{w_{M}}2r\lvert\\{w^{\sharp}>r\\}\cap(0,\tilde{x}_{t})\rvert\,dr.\end{split}$
(4.15)

We now claim that

$\lvert\\{w^{\sharp}>r\\}\cap(0,\tilde{x}_{t})\rvert\geq\lvert\\{w>r\\}\cap(0,\tilde{x}_{t})\rvert.$
(4.16)

Indeed, if we let

$\\{w>r\\}=(x_{r},y_{r}),\qquad\qquad\\{w^{\sharp}>r\\}=(x^{*}_{r},y^{*}_{r}),$

then (4.7) gives

$x^{\ast}_{r}=-\int_{0}^{r}\eta^{\prime}_{\ast,1}(s)\,ds\leq-\int_{0}^{r}\eta^{\prime}_{1}(s)\,ds=x_{r},$

while the equi-misurability of $w$ and $w^{\sharp}$ gives

$y^{\ast}_{r}=(y^{\ast}_{r}-x^{\ast}_{r})+x^{\ast}_{r}=(y_{r}-x_{r})+x^{\ast}_{r}\leq
y_{r}.$

Therefore we get

$\begin{split}\lvert\\{w^{\sharp}>r\\}\cap(0,\tilde{x}_{t})\rvert=\lvert\\{w>r\\}\cap(0,\tilde{x}_{t})\rvert\qquad\qquad\text{if
}y_{r}\leq\tilde{x}_{t},\\\\[5.0pt]
\lvert\\{w^{\sharp}>r\\}\cap(0,\tilde{x}_{t})\rvert>\lvert\\{w>r\\}\cap(0,\tilde{x}_{t})\rvert\qquad\qquad\text{if
}y_{r}>\tilde{x}_{t},\end{split}$

thus the claim is proved. Finally, joining (4.14), (4.15), and (4.16), we have
that

$\int_{D^{\ast}_{t}}g_{0}(x)\,dx\geq\int_{D_{t}}g(x)\,dx,$ (4.17)

and the result follows from (4.17), and (4.13). ∎

We now turn our attention to the eigenvalue problem.

###### Alternative proof of Theorem 1.1.

Let $h\in\mathcal{P}_{1}$, by Section 4 we have that

$\sigma_{1}(h)=\min\Set{\dfrac{\displaystyle\int_{0}^{1}(\varphi^{\prime})^{2}\,dx}{\displaystyle\int_{0}^{1}\frac{\varphi^{2}}{h}\,dx}\colon\,\varphi\in
H^{1}_{0}(0,1)}.$ (4.18)

Let $w$ be a minimizer in (4.18), then by Section 4, and Section 4, we have

$\sigma_{1}(h)=\frac{\displaystyle\int_{0}^{1}(w^{\prime})^{2}}{\displaystyle\int_{0}^{1}\frac{w^{2}}{h}}\geq\frac{\displaystyle{\int_{0}^{1}((w^{\sharp})^{\prime})^{2}}}{\displaystyle{\int_{0}^{1}\frac{(w^{\sharp})^{2}}{h^{\ast}}}}\geq\sigma_{1}(h^{\ast}).$
(4.19)

By Section 3, and Subsection 2.1, we have that the minimum of $\sigma_{1}$ is
a triangle $T_{x_{0}}$ for some $x_{0}\in[0,1]$. Let $h=T_{x_{0}}$, then
$h_{*}=T_{1}$ and, from (4.19), we have

$\sigma_{1}(T_{x_{0}})\geq\sigma_{1}(T_{1}),$

which concludes the proof. ∎

##  5 Ratio $\mu/\sigma$

In this section, we prove Theorem 1.2 and Theorem 1.3. We begin by defining an
operator $\mathcal{G}$ on $\mathcal{P}$ as follows: let $h\in\mathcal{P}$, and
let

$H(x)=\dfrac{1}{\int_{0}^{1}h(t)\,dt}\int_{0}^{x}h(t)\,dt;$ (5.1)

we notice that $H$ is a strictly increasing function such that $H(0)=0$ and
$H(1)=1$. We then define

$\mathcal{G}(h)(x)=h^{2}(H^{-1}(x)).$

######  Lemma .

Let $h\in\mathcal{P}$. Then $\mathcal{G}(h)\in\mathcal{P}$, and the map

$\mathcal{G}:\mathcal{P}\to\mathcal{P}$

is invertible.

###### Proof.

Since $h\in\mathcal{P}$, then $h^{\prime}$ is defined a.e. in $[0,1]$, and
$h^{\prime}$ is decreasing. We also have that $H^{-1}$ is a locally Lipschitz
function and

$\frac{d}{dx}H^{-1}(x)=\frac{1}{h(H^{-1}(x))}\int_{0}^{1}h(t)\,dt.$ (5.2)

Therefore, $\mathcal{G}(h)$ is a.e. differentiable and

$\frac{d}{dx}\mathcal{G}(h)(x)=2\alpha h^{\prime}(H^{-1}(x)),$

where

$\alpha=\int_{0}^{1}h(t)\,dt.$

Since $H^{-1}$ is an increasing function and $h^{\prime}$ is decreasing, then
$\mathcal{G}(h)$ is a concave function, and $\mathcal{G}(h)\in\mathcal{P}$.

Let $k\in\mathcal{P}$ and define

$K(x)=\dfrac{1}{\int_{0}^{1}\frac{1}{\sqrt{k(t)}}\,dt}\int_{0}^{x}\frac{1}{\sqrt{k(t)}}\,dt,$
(5.3)

then we want to prove that

$\sqrt{k(K^{-1}(x))}=\mathcal{G}^{-1}(k)(x).$ (5.4)

First we prove that $\sqrt{k\circ K^{-1}}\in\mathcal{P}$. By direct
computation,

$\frac{d}{dx}\sqrt{k(K^{-1}(x))}=\frac{\beta
k^{\prime}(K^{-1}(x))}{2k(K^{-1}(x))},$

where

$\beta=\int_{0}^{1}\frac{1}{\sqrt{k(t)}}\,dt.$

This proves that $\sqrt{k\circ K^{-1}}$ is concave, since $K^{-1}$ is
increasing and $h^{\prime}/h$ is decreasing because of the concavity of $h$.
On the other hand, to prove (5.4), we observe that with a change of variables
we get

$\int_{0}^{x}\sqrt{k(K^{-1}(t))}\,dt=\frac{K(x)}{\int_{0}^{1}\frac{1}{\sqrt{k(t)}}\,dt},$

and by definition of $\mathcal{G}$ we get

$\mathcal{G}\left(\sqrt{k\circ K^{-1}}\right)(x)=k(x).$

∎

We now prove that $\mathcal{G}$ is the operator in Theorem 1.2.

###### Proof of Theorem 1.2.

Let $v\in H^{1}(0,1)$ be a function such that

$\int_{0}^{1}v(t)h(t)\,dt=0,$

and let $H$ denote the integral function defined in (5.1). The change of
variables $H(t)=s$ yields

$\left(\int_{0}^{1}h(t)\,dt\right)\int_{0}^{1}v(t)\,h(t)\,dt=\int_{0}^{1}v(H^{-1}(s))\,ds,$
$\left(\int_{0}^{1}h(t)\,dt\right)\int_{0}^{1}(v^{\prime})^{2}(t)\,h(t)\,dt=\int_{0}^{1}(v^{\prime})^{2}(H^{-1}(s))\,ds,$

and

$\left(\int_{0}^{1}h(t)\,dt\right)\int_{0}^{1}v^{2}(t)\,h(t)\,dt=\int_{0}^{1}v^{2}(H^{-1}(s))\,ds.$

Let $w(x)=v(H^{-1}(x))$, then by (5.2),

$w^{\prime}(x)=\left(\int_{0}^{1}h(t)\,dt\right)\,v^{\prime}(H^{-1}(x))\,\bigl{(}\mathcal{G}(h)(x)\bigr{)}^{-\frac{1}{2}}.$

Hence,

$\frac{\displaystyle\int_{0}^{1}(v^{\prime})^{2}(t)\,h(t)\,dt}{\displaystyle\int_{0}^{1}v^{2}(t)\,h(t)\,dt}=\left(\int_{0}^{1}h(t)\,dt\right)^{-2}\frac{\displaystyle\int_{0}^{1}(w^{\prime})^{2}(t)\,\mathcal{G}(h)(t)\,dt}{\displaystyle\int_{0}^{1}w^{2}(t)\,dt}.$

Choosing $v=v_{\mu}$ to be the eigenfunction of $\mu_{1}(h)$, then we get

$\mu_{1}(h)\geq\left(\int_{0}^{1}h(t)\,dt\right)^{-2}\sigma_{1}(h).$

On the other hand, choosing $w=w_{\sigma}$ to be the eigenfunction of
$\sigma_{1}(\mathcal{G}(h))$, we get

$\mu_{1}(h)\leq\left(\int_{0}^{1}h(t)\,dt\right)^{-2}\sigma(\mathcal{G}(h)),$

which gives (1.6).

Let $k\in\mathcal{P}$ and let $K$ be the integral function defined in (5.3).
If we evaluate the integral on the right-hand side by means of the change of
variables $t=K(s)$, we finally get

$\int_{0}^{1}h(t)\,dt=\left(\int_{0}^{1}\frac{1}{\sqrt{k(t)}}\,dt\right)^{-1},$

which gives (1.7). ∎

The following punctual estimate will be crucial.

######  Proposition .

Let $h\in\mathcal{P}$. Then

$\left(\int_{0}^{1}h(t)\,dt\right)^{-1}\mathcal{G}(h)(x)\leq 2h(x).$

###### Proof.

Up to rescaling $h$, we can assume without loss of generality that
$h\in\mathcal{P}_{1}$. Notice, in addition, that if $h\equiv 1$, then the
proof is trivial. Therefore, let $h\in\mathcal{P}_{1}$ and $h\neq 1$, and
define

$H(x)=\int_{0}^{x}h(t)\,dt.$

We claim that there exists a unique $\bar{x}\in[0,1]$ such that

$\displaystyle H(x)\leq x$ $\displaystyle x\in[0,\bar{x}],$ (5.5)
$\displaystyle H(x)\geq x$ $\displaystyle x\in[\bar{x},1].$

Indeed, if we denote by

$f(x)=H(x)-x,$

then, by the concavity of $h$ and the integral constraint, we have that the
equation $h=1$ admits at most two solutions ($h$ cannot be equal to 1 in an
entire interval, otherwise the concavity would give $\lVert h\rVert_{1}<1$).
Therefore, we have that there exist two points $x_{1}\in[0,1)$, and
$x_{2}\in(0,1]$ such that

$\displaystyle f^{\prime}(x)<0\qquad\qquad$ $\displaystyle
x\in[0,x_{1})\cup(x_{2},1],$ $\displaystyle f^{\prime}(x)>0\qquad\qquad$
$\displaystyle x\in(x_{1},x_{2}).$

Finally, noticing that $f(0)=f(1)=0$, then we have that there exists a unique
zero $\bar{x}$ of $f$ in the interval $[x_{1},x_{2}]$, thus the claim is
proved. In particular, we have that

$\displaystyle H^{-1}(x)\geq x\qquad\qquad$ $\displaystyle x\in[0,\bar{x}],$
$\displaystyle H^{-1}(x)\leq x\qquad\qquad$ $\displaystyle x\in[\bar{x},1],$

and $h(\bar{x})>1$.

These estimates allow us to compare the derivatives of $\mathcal{G}(h)$ and
$h$. Denoting by $g(x)=\mathcal{G}(h)(x)$, we have that

$\displaystyle g^{\prime}(x)=2h^{\prime}(H^{-1}(x))\leq
2h^{\prime}(x)\qquad\qquad$ $\displaystyle x\in[0,\bar{x}],$ (5.6)
$\displaystyle g^{\prime}(x)=2h^{\prime}(H^{-1}(x))\geq
2h^{\prime}(x)\qquad\qquad$ $\displaystyle x\in[\bar{x},1].$ (5.7)

We recall that, as in (2.4), the concavity of $h$ ensures that

${\lVert h\rVert_{\infty}}\leq 2.$

Therefore, we get

$g(0)=h^{2}(0)\leq 2h(0),$

and by (5.6),

$g(x)\leq 2h(x)\qquad\qquad x\in[0,\bar{x}].$

Analogously,

$g(1)=h^{2}(1)\leq 2h(1),$

and, by (5.7),

$g(x)\leq 2h(x)\qquad\qquad x\in[\bar{x},1].$

∎

######  Proposition .

Let $h\in\mathcal{P}_{1}$.

$\mathcal{G}(h)=2h$

if and only if $h=T_{x_{0}}$ for some $x_{0}\in[0,1]$.

###### Proof.

By direct computation, one can prove that if $h=T_{x_{0}}$ for some
$x_{0}\in[0,1]$, then

$h^{2}(x)=2h(H(x)).$

Let us now assume that $\mathcal{G}(h)=2h$. Notice that, if $y\in[0,1]$ is a
fixed point of the integral function $H$, then

$h^{2}(y)=h^{2}(H^{-1}(y))=\mathcal{G}(h)(y)=2h(y),$ (5.8)

so that either $h(y)=0$ or $h(y)=2$. In particular, if $h(y)=2$, then by the
concavity of $h$, we have that

$h(x)=T_{y}(x)\qquad\forall\,x\in[0,1].$

Since $0$ and $1$ are always fixed points of $H$, if either $h(0)=2$ or
$h(1)=2$ the assertion is proved. Therefore, let us assume that

$h(0)=0=h(1),$

then the equation $h=1$ admits at least two distinct solutions
$0<x_{1}<x_{2}<1$ and, arguing as in the proof of Section 5, we have that
there exists a fixed point $\bar{x}\in[x_{1},x_{2}]$ for the function $H$ and,
by (5.8), necessarily $h(\bar{x})=2$ and $h=T_{\bar{x}}$. ∎

######  Proposition .

Let $h\in\mathcal{P}_{1}$, then

$\frac{\mu_{1}(h)}{\sigma_{1}(h)}\leq 2$ (5.9)

and the equality holds if and only if $h=T_{x_{0}}$ for some $x_{0}\in[0,1]$.

###### Proof.

Let $w$ be an eigenfunction for $\sigma_{1}(h)$. Using Theorem Theorem 1.2,
Section 5, and the variational characterization of
$\sigma_{1}(\mathcal{G}(h))$, we obtain

$\mu_{1}(h)=\sigma_{1}(\mathcal{G}(h))\leq\frac{\displaystyle\int_{0}^{1}(w^{\prime})^{2}\mathcal{G}(h)\,dx}{\displaystyle\int_{0}^{1}w^{2}\,dx}\leq\frac{\displaystyle
2\int_{0}^{1}(w^{\prime})^{2}h\,dx}{\displaystyle\int_{0}^{1}w^{2}\,dx}=2\sigma_{1}(h),$
(5.10)

thus proving (5.9). Assume now that for some $h\in\mathcal{P}_{1}$ equality
holds, then by (5.10) we have

$\int_{0}^{1}(w^{\prime})^{2}(\mathcal{G}(h)-2h)\,dx=0.$ (5.11)

Since $\mathcal{G}(h)\leq 2h$, then (5.11) yields $\mathcal{G}(h)=2h$, and
Section 5 ensures that $h=T_{x_{0}}$ for some $x_{0}\in[0,1]$. ∎

######  Remark .

Since it is not possible in general to have that $\mathcal{G}(h)\geq h$, then
the same argument cannot be used for the lower bound

$\frac{\mu_{1}(h)}{\sigma_{1}(h)}\geq 1.$

For instance, let

$h(x)=\frac{1}{2}+x,$

then $\mathcal{G}(h)(0)=h^{2}(0)<h(0)$, while
$\mathcal{G}(h)(1)=h^{2}(1)>h(1)$.

Here we prove the lower bound in Theorem 1.3 in the symmetric case.

######  Proposition .

Let $h\in\mathcal{P}_{1}$ such that $h(1-x)=h(x)$ for all $x\in[0,1]$. Then

$\frac{\mu_{1}(h)}{\sigma_{1}(h)}\geq 1.$ (5.12)

###### Proof.

Let $g=\mathcal{G}(h)$. By the variational characterization (4.18) of
$\sigma_{1}$, and Theorem 1.2, we can find a function $w\in H^{2}(0,1)$,
symmetric with respect to $x=1/2$, such that

$\mu_{1}(h)=\sigma_{1}(g)=\frac{\displaystyle\int_{0}^{1}(w^{\prime})^{2}(x)\,dx}{\displaystyle\int_{0}^{1}\frac{w^{2}(x)}{g(x)}\,dx},$
(5.13)

and $w$ solves the problem

$\begin{cases}\displaystyle{-w^{\prime\prime}(x)=\frac{\sigma_{1}(h)}{h(x)}w(x)}\qquad\qquad
x\in(0,1),\\\\[7.0pt] w(0)=w(1)=0.\end{cases}$

We can choose $w$ to be positive and concave, so that

$\displaystyle w^{\prime}(x)\geq 0$ $\displaystyle\text{in
}\left(0,\frac{1}{2}\right),$ (5.14) $\displaystyle w^{\prime}(x)\leq 0$
$\displaystyle\text{in }\left(\frac{1}{2},1\right).$

Moreover, by the variational characterization (4.18), we get

$\sigma_{1}(h)\leq\frac{\displaystyle\int_{0}^{1}(w^{\prime})^{2}(x)\,dx}{\displaystyle\int_{0}^{1}\frac{w^{2}(x)}{h(x)}\,dx},$
(5.15)

and then, joining (5.13) and (5.15), we get

$\frac{\mu_{1}(h)}{\sigma_{1}(h)}\geq\frac{\displaystyle\int_{0}^{1}\frac{w^{2}(x)}{h(x)}\,dx}{\displaystyle\int_{0}^{1}\frac{w^{2}(x)}{g(x)}\,dx}.$

To prove (5.12) it is sufficient to prove that

$\int_{0}^{1}\frac{w^{2}(x)}{g(x)}\,dx\leq\int_{0}^{1}\frac{w^{2}(x)}{h(x)}\,dx.$

We now compute the left-hand side by means of the change of variables
$x=H(y)$, where

$H(y)=\int_{0}^{y}h(t)\,dt,$

so that

$\int_{0}^{1}\frac{w^{2}(x)}{g(x)}\,dx=\int_{0}^{1}\frac{w^{2}(H(y))}{h(y)}\,dy.$

We now notice that the symmetry of $h$ gives (5.5) with $\bar{x}=1/2$, namely

$\displaystyle H(y)\leq y$ $\displaystyle x\in\left[0,\frac{1}{2}\right],$
(5.16) $\displaystyle H(y)\geq y$ $\displaystyle
x\in\left[\frac{1}{2},1\right].$

Finally, joining (5.16), and (5.14), we have

$\int_{0}^{1}\frac{w^{2}(x)}{g(x)}\,dx=\int_{0}^{1}\frac{w^{2}(H(y))}{h(y)}\,dy\leq\int_{0}^{1}\frac{w^{2}(y)}{h(y)}\,dy,$

which concludes the proof. ∎

###### Proof of Theorem 1.3.

The result follows from Section 5 and Section 5 ∎

### Acknowledgements

We would like to thank Carlo Nitsch and Cristina Trombetti for the valuable
advice that helped us to achieve these results.

The three authors were partially supported by Gruppo Nazionale per l’Analisi
Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto
Nazionale di Alta Matematica (INdAM).

### Competing Interests

The authors report there are no competing interests to declare.

## References

  * [1] L. Ambrosio, N. Fusco and D. Pallara  “Functions of bounded variation and free discontinuity problems”, Oxford Mathematical Monographs  The Clarendon Press, Oxford University Press, New York, 2000, pp. xviii+434
  * [2] D. Bucur, A. Henrot and M. Michetti  “Asymptotic behaviour of the Steklov spectrum on dumbbell domains”  In _Comm. Partial Differential Equations_ 46.2, 2021, pp. 362–393  DOI: 10.1080/03605302.2020.1840587
  * [3] V. Ferone, C. Nitsch and C. Trombetti  “A remark on optimal weighted Poincaré inequalities for convex domains”  In _Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl._ 23.4, 2012, pp. 467–475  DOI: 10.4171/RLM/640
  * [4] A. Girouard, A. Henrot and J. Lagacé  “From Steklov to Neumann via homogenisation”  In _Arch. Ration. Mech. Anal._ 239.2, 2021, pp. 981–1023  DOI: 10.1007/s00205-020-01588-2
  * [5] A. Girouard, M. Karpukhin and J. Lagacé  “Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems”  In _Geom. Funct. Anal._ 31.3, 2021, pp. 513–561  DOI: 10.1007/s00039-021-00573-5
  * [6] A. Henrot and M. Michetti  “A comparison between Neumann and Steklov eigenvalues”  In _J. Spectr. Theory_ 12.4, 2022, pp. 1405–1442  DOI: 10.4171/jst/429
  * [7] A. Kufner, Lars-E. Persson and N. Samko  “Weighted inequalities of Hardy type”  World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017, pp. xx+459  DOI: 10.1142/10052
  * [8] P.D. Lamberti and L. Provenzano  “Viewing the Steklov eigenvalues of the Laplace operator as critical Neumann eigenvalues”  In _Current trends in analysis and its applications_ , Trends Math.  Birkhäuser/Springer, Cham, 2015, pp. 171–178
  * [9] F. Maggi  “Sets of finite perimeter and geometric variational problems” An introduction to geometric measure theory 135, Cambridge Studies in Advanced Mathematics  Cambridge University Press, Cambridge, 2012, pp. xx+454  DOI: 10.1017/CBO9781139108133
  * [10] B. Opic and A. Kufner  “Hardy-type inequalities” 219, Pitman Research Notes in Mathematics Series  Longman Scientific & Technical, Harlow, 1990, pp. xii+333
  * [11] L.. Payne and H.. Weinberger  “An optimal Poincaré inequality for convex domains”  In _Arch. Rational Mech. Anal._ 5, 1960, pp. 286–292  DOI: 10.1007/BF00252910
  * [12] B.A. Troesch  “An isoperimetric sloshing problem”  In _Communications on Pure and Applied Mathematics_ 18.1-2 Wiley Online Library, 1965, pp. 319–338

Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli
studi di Napoli Federico II, Via Cintia, Complesso Universitario Monte S.
Angelo, 80126 Napoli, Italy.

E-mail address, P. Acampora<EMAIL_ADDRESS>
E-mail address, V. Amato<EMAIL_ADDRESS>
Mathematical and Physical Sciences for Advanced Materials and Technologies,
Scuola Superiore Meridionale, Largo San Marcellino 10, 80126, Napoli, Italy.

E-mail address, E. Cristoforoni<EMAIL_ADDRESS>


# Parametric Continued Fractions for $\pi^{2}$, $\zeta(3)$, and other
Constants

Henri Cohen  Université de Bordeaux, LFANT,
IMB, U.M.R. 5251 du C.N.R.S., 351 Cours de la Libération
33405 Talence Cedex, FRANCE

###### Abstract

We give an extensive list of parametrized _families_ of polynomial continued
fractions of _smallest possible degrees_ for $\pi^{2}$ and $\zeta(3)$, and
mention similar results for other constants.

##  1 Introduction

One can find in the literature a huge number of continued fractions
(abbreviated CFs) for interesting constants and functions. The purpose of this
paper is to give possibly _experimental_ results showing that there often
exist _families_ of such CFs, usually depending on two, three, and even four
parameters. In many cases these experimental results can be _proved_ , usually
because either $p_{n}/q_{n}$ or $p_{n}/q_{n}-p_{n-1}/q_{n-1}$ have an explicit
expression. In other cases, I have not bothered to try to prove the
experimental results, although it may not be difficult. Thus, when I speak of
results, they should be considered experimental, but with considerable
evidence. I would appreciate receiving any additional data of the same kind.

Notation: a CF will be written as a pair of sequences $(a_{n},b_{n})_{n\geq
0}$ representing the CF
$a_{0}+b_{0}/(a_{1}+b_{1}/(a_{2}+b_{2}/(a_{3}+\cdots)))$ (warning: note this
precise convention, other authors have different notation). All the CFs that
we will encounter will be _convergent_ : if as usual
$p_{n}/q_{n}=a_{0}+b_{0}/(a_{1}+\cdots+b_{n-1}/a_{n})$, this means that
$z=\lim_{n\to\infty}p_{n}/q_{n}$ exists, and when we speak of the speed of
convergence, we mean the behavior of $z-p_{n}/q_{n}$.

Note that we always assume that $b_{n}\neq 0$ for all $n$, but we allow a
finite number of $a_{n}$ to vanish.

###### Definition 1.1

  1. (1)

Let $(a_{n})_{n\geq 0}$ be a sequence (always assumed to be of rational
numbers, and usually of integers). We say that it is ultimately polynomial if
there exists a polynomial denoted $a(x)$ (by abuse of notation) with rational
coefficients such that $a_{n}=a(n)$ and $a_{n}\neq 0$ for $n$ sufficiently
large.

  2. (2)

We say that the CF $(a_{n},b_{n})_{n\geq 0}$ is of _polynomial type_ if the
sequences $a_{n}$ and $b_{n}$ are ultimately polynomial.

As a practical notation, if $a_{n}=a(n)$ for $n\geq n_{0}$, we will denote the
sequence $(a_{n})_{n\geq 0}$ by the $(n_{0}+1)$-component vector
$A=(a_{0},a_{1},\dotsc,a_{n_{0}-1},a(n))$, where the last entry is a
polynomial. Of course this is not unique: we also have for instance
$A=(a_{0},a_{1},\dotsc,a_{n_{0}-1},a_{n_{0}},a(n))$ with $a_{n_{0}}=a(n_{0})$.

Thus a CF will be denoted by a pair of such vectors $A$ and $B$ representing
$(a_{n})$ and $(b_{n})$, and if the CF converges to some number $z$, we will
write $z=(a_{n},b_{n})$.

###### Definition 1.2

We recall that two CFs $(a_{n},b_{n})$ and $(a^{\prime}_{n},b^{\prime}_{n})$
are said to be _equivalent_ if there exists an integer $k$ such that
$(a^{\prime}_{n},b^{\prime}_{n})=(a_{n+k},b_{n+k})$ for $n$ sufficiently
large, i.e., if the _tails_ of the CFs are ultimately the same.

Since we assume everything rational, if we set $z=(a_{n},b_{n})$ and
$z^{\prime}=(a^{\prime}_{n},b^{\prime}_{n})$, the CFs for $z$ and $z^{\prime}$
are equivalent if and only if there exists a Möbius transformation with
integer coefficients sending $z$ to $z^{\prime}$, in other words
$z^{\prime}=(Az+B)/(Cz+D)$ for integers $A$, $B$, $C$, $D$ with $AD-BC\neq 0$.

We can now give our final definition:

###### Definition 1.3

We say that a pair of polynomials $(A(x),B(x))$ with rational coefficients
_represents_ a constant $z$ if there exists a CF $(a_{n},b_{n})$ converging to
$z$ such that $a_{n}=A(n)$ and $b_{n}=B(n)$ for $n$ sufficiently large, and in
that case we simply write by abuse of notation $z=(A(n),B(n))$.

Because of what we have said above, this is almost, but not quite, equivalent
to the fact that the CF $(A(n),B(n))_{n\geq 0}$ converges to a Möbius
transform of $z$ (always assumed with integer coefficients): indeed, some
values of $A(n)$ and/or $B(n)$ may vanish, which kills this property. But if
we avoid these vanishing values by changing the corresponding coefficients of
the CF to any nonzero value, this is indeed true.

Our goal is, given some interesting constant $z$, to find all (or at least
many) pairs $(A,B)$ representing $z$, subject to suitable restrictions. In
particular, recall that if $(a_{n},b_{n})_{n\geq 0}$ is a CF converging to $z$
and $r_{n}$ is any sequence with $r_{0}=1$ and $r_{n}\neq 0$ for all $n$, the
CF $(a_{n}r_{n},b_{n}r_{n}r_{n+1})_{n\geq 0}$ also converges to $z$. We will
thus restrict to polynomials $(A,B)$ of smallest possible degrees.

To make this more precise, we introduce the following definitions:

###### Definition 1.4

We say that $z\in{\mathbb{C}}$ is a _rational period of degree $k$_ if it is
equal to the sum of a convergent series of the form $\sum_{n\geq
1}\chi(n)f(n)$, where $\chi$ is a periodic arithmetic function taking
_rational_ values, and $f\in{\mathbb{Q}}(x)$ is a rational function with
rational coefficients whose denominator is of degree $k$.

Note that this definition is _not_ compatible with the definition of periods
as given by Kontsevitch–Zagier [3]. Note also that a ${\mathbb{Q}}$-linear
combination of rational periods of degree $\leq k$ is again a rational period
of degree $\leq k$.

Examples of periods of degree 1: $\pi$, $\pi/\sqrt{d}$, $\log(2)$,
$\sqrt{2}\log(1+\sqrt{2})$, $\sqrt{3}\log(2+\sqrt{3})$,
$\sqrt{5}\log((1+\sqrt{5})/2)$.

Examples of periods of degree 2: $\pi^{2}$, $\pi^{2}/\sqrt{d}$,
$G=L(\chi_{-4},2)$ (Catalan’s constant), $G_{3}=L(\chi_{-3},2)$.

Examples of periods of degree 3: $\pi^{3}$, $\pi^{3}/\sqrt{d}$, $\zeta(3)$.

By using Euler’s transformation of series it is clear that a period has a CF
expansion where $a(n)$ and $b(n)$ are polynomials with rational coefficients
for $n$ large. However this is usually uninteresting, or at least not any more
interesting than the series itself. Thus, really interesting CFs are those for
which the degrees of the polynomials are reasonably small, specifically
$\deg(a(n))\leq k$ and $\deg(b(n))\leq 2k$ for $n$ sufficiently large (we will
say that it has _bidegree_ $\leq(k,2k)$). Trivial examples: $S^{+}=\sum_{n\geq
1}1/P(n)$ and $S^{-}=\sum_{n\geq 1}(-1)^{n-1}/P(n)$ with $\deg(P)=k\geq 2$
have the CFs $S^{+}=((0,P(n)+P(n-1)),(1,-P(n)^{2}))$ and
$S^{-}=((0,P(n)-P(n-1)),(1,P(n)^{2}))$ by Euler, of the required degrees.

Note that if we apply Euler’s transformation to $S=\sum_{n\geq
1}(n+1)/n^{3}=\zeta(3)+\zeta(2)$ which is a period of degree $3$, we obtain a
CF of bidegree $(4,8)$. However, there does exist a CF of bidegree $(3,6)$,
for instance we have the (easily proved) CF:

$\zeta(3)+\zeta(2)=((3,2n^{3}+2n^{2}+3n+1),(-1,-n(n+1)^{5}))\;.$

For most of this paper, we will focus on the cases $z=\pi^{2}$ (or
equivalently $z=\zeta(2)=\pi^{2}/6$) and $z=\zeta(3)$, and attempt to classify
_all_ representations of $\pi^{2}$ with bidegree $\leq(2,4)$, and of
$\zeta(3)$ with bidegree $\leq(3,6)$. A completely analogous study can be made
for other periods such as those mentioned above, we will briefly mention this
in the last section.

Note that, contrary to [1], we are interested in CFs for “pure” periods such
as those mentioned above, and not in linear combinations, which is
considerably more restrictive since one needs to get rid of “parasitic” terms
(for instance $\log(2)$ for $\pi^{2}$, or $\log(2)$ and $\zeta(2)$ for
$\zeta(3)$, see below).

##  2 CFs for $\pi^{2}$: Convergence in $1/n^{P}$, $P\in{\mathbb{Z}}_{\geq 1}$

As far as the author is aware, up to equivalence there exist exactly nine
types of convergence of polynomial continued fractions for $\pi^{2}$ with
bidegree $\leq(2,4)$: two families with convergence in $1/n^{P}$ with
$P\in{\mathbb{Z}}$, one with convergence in $1/n^{P}$ with $P\in
1/2+{\mathbb{Z}}$, one with convergence in $(-1)^{n}/n^{P}$, one with
convergence $1/(E^{n}n^{P})$ for each of $E=4/3$, $2$, $4$, and $-8$, and
finally the unique CF of Apéry with convergence in $1/E^{n}$ for
$E=-((1+\sqrt{5})/2)^{10}$. We consider each of these families in turn.

###  2.1 First Family

Using Euler’s transformation of series into CFs applied to
$\pi^{2}/6=\sum_{n\geq 1}1/n^{2}$, we see that $\pi^{2}/6$ is given by the CF
$((0,2n^{2}-2n+1),(1,-n^{4}))$, with speed of convergence identical to that of
the series, in $1/n$, which we write in abbreviated form as explained above
where we do not give the initial values of $a_{n}$ and $b_{n}$

$\pi^{2}=(2n^{2}-2n+1,-n^{4})\;.$

Remark: even though this notation means that there can be a number of initial
values of $a_{n}$ and $b_{n}$ which are not given by the general polynomial
formulas, in most of the examples that we will see in this paper, only $a_{0}$
and $b_{0}$ would need to be specified (for instance here $a_{0}=0$ and
$b_{0}=1$), the other $a_{n}$ and $b_{n}$ for $n\geq 1$ are given by the
polynomials.

If we apply the Bauer–Muir transformation, we check that for any nonnegative
integer $k\geq 0$ we also have $\pi^{2}=(2n^{2}-2n+k^{2}+k+1,-n^{4})$.

This naturally leads us to _search numerically_ for other representations of
the form $\pi^{2}=(A(n),B(n))$, with $A(n)=2n^{2}+a_{1}n+a_{0}$ and $B(n)$ of
degree $4$ with leading coefficient $-1$. We speed up the search by assuming
that $B$ is a product of linear factors with integral coefficients (you are
welcome to try other types of polynomials, but I am pretty sure that you will
not find others), and since $(A(n+j),B(n+j))$ is equivalent to $(A(n),B(n))$,
we can assume that $B(n)=-n(n+u_{1})(n+u_{2})(n+u_{3})$ with $0\leq u_{1}\leq
u_{2}\leq u_{3}$.

We find a _very large_ number of such representations, from which one can
easily extract one and two-parameter families, less easily three-parameter
families, and finally with considerable difficulty, a single four-parameter
family to rule them all: $\pi^{2}=(A(n),B(n))$ with

$\displaystyle A(n)$ $\displaystyle=2n^{2}+(3u+v+w-2)n$
$\displaystyle\phantom{=}+k^{2}+u^{2}+uk+vk+wk+uv+uw+vw+k-u+1$ $\displaystyle
B(n)$ $\displaystyle=-n(n+u)(n+u+v)(n+u+w)\;,$

where $u$, $v$, $w$, and $k$ are nonnegative integer parameters, with speed of
convergence in $C/n^{u+v+w+2k+1}$ for some constant $C$.

An extensive additional search seems to show that there are no additional
sporadic representations. However, we will see in Section 3 that it is also
possible to choose $v$ or $w$ (but not both) in $1/2+{\mathbb{Z}}$.

Proof: in the present case, it is possible, not only to prove that the four-
parameter family above does represent $\pi^{2}$, but also to explain where it
originates.

Consider the series $S=\sum_{n\geq 1}1/\prod_{0\leq i\leq m-1}(n+i)^{e_{i}}$,
where $e_{i}=1$ if $0\leq i\leq m_{1}-1$ or $m_{1}+m_{2}\leq i\leq m-1$, and
$e_{i}=2$ if $m_{1}\leq i\leq m_{1}+m_{2}-1$, where $m_{1}\geq 0$, $m_{2}\geq
1$, and $m\geq m_{1}+m_{2}$. By expanding into partial fractions it is
immediate to show that the sum $S$ satisfies $S=a\pi^{2}+b$ with $a$, $b$
rational and $a\neq 0$. On the other hand, applying Euler’s transformation of
$S$ into a CF, and simplifying by suitable factors, it is not difficult to
show that $S$ (hence $\pi^{2}$) is represented by the CF $(A(n),B(n))$, with

$\displaystyle A(n)$
$\displaystyle=(n-1)(n+m_{1}-1)+(n+m_{1}+m_{2}-1)(n+m-1)\text{\quad and\quad}$
$\displaystyle B(n)$ $\displaystyle=-n(n+m_{1})(n+m_{1}+m_{2}-1)(n+m-1)\;.$

This gives the three-parameter family above when $k=0$, and the case of
general $k$ is obtained by successive Bauer–Muir accelerations.

###  2.2 Second Family

We also have the modified series obtained by removing the Euler factor at $2$:
$\pi^{2}/8=\sum_{n\geq 1}1/(2n-1)^{2}$, giving the CF
$((0,1,8n^{2}-16n+10),(1,-(2n-1)^{4}))$ for $\pi^{2}/8$, that we write in
abbreviated form

$\pi^{2}=(8n^{2}-16n+10,-(2n-1)^{4})\;.$

All the work has been done for us in the first family: we simply replace $n$
by $n-1/2$ and multiply by suitable powers of $2$ to simplify the formulas,
giving the single four-parameter family $\pi^{2}=(A(n),B(n))$ with

$\displaystyle A(n)$ $\displaystyle=8n^{2}+4(3u+v+w-4)n$
$\displaystyle\phantom{=}+4(k^{2}+u^{2}+uk+vk+wk+uv+uw+vw)+4k-10u-2v-2w+10$
$\displaystyle B(n)$
$\displaystyle=-(2n-1)(2n-1+2u)(2n-1+2u+2v)(2n-1+2u+2w)\;,$

with the same speed of convergence in $C/n^{u+v+w+2k+1}$.

However, inspection shows that while $v$, $w$, and $k$ must still be
nonnegative, we can have $u$ negative as long as $u\geq-k-\min(v,w)$. Note
that here we _cannot_ choose $v$ or $w$ in $1/2+{\mathbb{Z}}$.

##  3 CFs for $\pi^{2}$: Convergence in $1/n^{P}$, $P\in
1/2+{\mathbb{Z}}_{\geq 0}$

We recall the following formula due to Euler:

$\sum_{n\geq
1}\dfrac{(2z)^{2n}}{n^{2}\binom{2n}{n}}=2\operatorname{asin}^{2}(z)\;,$

which we will use several times. Here, we chooose $z=1$ to obtain the series

$\sum_{n\geq 1}\dfrac{4^{n}}{n^{2}\binom{2n}{n}}=\dfrac{\pi^{2}}{2}\;,$

and using Euler’s transformation of series into CFs we see that $\pi^{2}/2$ is
given by the CF $\pi^{2}/2=((0,4n^{2}-5n+2),(2,-2n^{3}(2n-1)))$ with
convergence identical to that of the series in $\sqrt{\pi}/n^{1/2}$, which we
write in abbreviated form

$\pi^{2}=(4n^{2}-5n+2,-2n^{3}(2n-1))\;.$

If we apply the Bauer–Muir transformation, we check that for any nonnegative
integer $k\geq 0$ we also have $\pi^{2}=(4n^{2}-5n+2k^{2}+k+2,-2n^{3}(2n-1))$.

Here, we could do a new search, but luckily we have already done the work: it
is simply the first four-parameter family found above, but now with the
parameter $w$ being a half integer instead of an integer. We thus replace $w$
by $w-1/2$ and simplify, so we obtain the four-parameter family
$\pi^{2}=(A(n),B(n))$ with

$\displaystyle A(n)$ $\displaystyle=4n^{2}+(6u+2v+2w-5)n+2k^{2}$
$\displaystyle\phantom{=}+(2u+2v+2w+1)k+2u^{2}+(2v+2w-3)u+v(2w-1)+2$
$\displaystyle B(n)$ $\displaystyle=-2n(n+u)(n+u+v)(2n+2u+2w-1)\;,$

where $u$, $v$, and $k$ are nonnegative integer parameters and
$w\in{\mathbb{Z}}$, with speed of convergence in $C/n^{u+v+w+2k+1/2}$ for some
constant $C$.

The _proof_ of the validity of this family follows from the one given in
Section 2.1.

##  4 CFs for $\pi^{2}$: Convergence in $(-1)^{n}/n^{P}$

Using Euler’s transformation of series into CFs but now applied to
$\pi^{2}/12=\sum_{n\geq 1}(-1)^{n-1}/n^{2}$, we see that $\pi^{2}/12$ is given
by the CF $((0,2n-1),(1,n^{4}))$, with speed of convergence identical to that
of the series, in $(-1)^{n}/(2n^{2})$, which we write in abbreviated form

$\pi^{2}=(2n-1,n^{4})\;.$

If we apply the Bauer–Muir transformation, we immediately see that for any
_odd_ integer $k\geq 1$ we also have $\pi^{2}=(k(2n-1),n^{4})$. We easily
check that this is also true for even integers, so we have found a one-
parameter family. Now we search numerically for other representations of the
form $\pi^{2}=(A(n),B(n))$, with $A(n)$ of degree $1$ and $B(n)$ monic of
degree $4$, as before assumed to be a product of linear factors with integral
coefficients. Since $(-A(n),B(n))$ is easily seen to be equivalent to
$(A(n),B(n))$, we can also assume that the leading coefficient of $A(n)$ is
positive, so we set $A(n)=a_{1}n+a_{0}$ with $a_{1}\geq 1$ and
$B(n)=n(n+u_{1})(n+u_{2})(n+u_{3})$ with $0\leq u_{1}\leq u_{2}\leq u_{3}$. We
again find a large number of representations, and after some cleaning up, we
obtain a three-parameter family and four one-parameter families, and no
additional sporadic representations:

$\displaystyle\pi^{2}$
$\displaystyle=((k+1)(2n+2u+2v-1),n(n+u)(n+u+2v)(n+2u+2v))$
$\displaystyle=((2u^{2}+8u+7)n+(2u+3)(u^{2}+3u+1),n(n+u)(n+u+1)(n+2u+4))$
$\displaystyle=((2u^{2}+8u+7)n+(2u+3)(u+2)(u+3),n(n+u+3)(n+u+4)(n+2u+4))$
$\displaystyle=((8u+7)n+(24u^{2}+24u+3),n(n+2u)(n+4u+1)(n+6u+4))$
$\displaystyle=((8u+7)n+6(u+1)(4u+3),n(n+2u+3)(n+4u+4)(n+6u+4))\;,$

where $u$, $v$, and $k$ are nonnegative integer parameters, with speeds of
convergence $(-1)^{n}C/n^{2k+2}$, $(-1)^{n}C/n^{2u^{2}+8u+7}$,
$(-1)^{n}C/n^{2u^{2}+8u+7}$, $(-1)^{n}C/n^{8u+7}$ and $(-1)^{n}C/n^{8u+7}$
respectively, for some constants $C$.

Proof: Although similar to the proof given in Section 2.1 the proof of the
above is both more complicated and more interesting since we will need to
generalize the construction. Here we consider the series

$S=\sum_{n\geq 1}(-1)^{n-1}/\prod_{0\leq i\leq m-1}(n+i)^{e_{i}}\;,$

with the same $e_{i}$ as above. Once again we can simplify the result given by
Euler’s transformation of $S$ into a CF, and we obtain that $S$ is represented
by the CF $(A(n),B(n))$ with

$\displaystyle A(n)$
$\displaystyle=-(n-1)(n+m_{1}-1)+(n+m_{1}+m_{2}-1)(n+m-1)\text{\quad
and\quad}$ $\displaystyle B(n)$
$\displaystyle=n(n+m_{1})(n+m_{1}+m_{2}-1)(n+m-1)\;.$

The problem is that the sum $S$ is not anymore of the form $a\pi^{2}+b$, but
of the form $a\pi^{2}+c\log(2)+b$. Thus, if we want a CF representing
$\pi^{2}$ itself, we need $c=0$. It is not difficult to show that this is
indeed the case when $(m_{1},m_{2},m_{3}):=(m_{1},m_{2},m-m_{1}-m_{2})$ is of
one of the following three shapes: $(m_{1},m_{2},m_{3})=(m_{1},2u+1,m_{1})$ or
$(2u,2u+2,2u+3)$, or $(2u+3,2u+2,2u)$. It should not be difficult to prove
that there are no other cases, but I have not done so. In any event, the first
shape followed by Bauer–Muir accelerations leads to the three-parameter
family, and the other two shapes lead to the last two one-parameter families.

The first two one-parameter families do _not_ come from this construction, but
from a more general one. We first note the following result, whose easy proof
is left to the reader

###### Proposition 4.1

Let $c(x)\in{\mathbb{Q}}[x]$ be a polynomial with $\deg(c)\geq 2$, let $z=\pm
1$, let $P\in{\mathbb{Q}}[x]$ be an irreducible polynomial which divides
$c(x)P(x+1)+zc(x-1)P(x-1)$, let $R(x)=(c(x)P(x+1)+zc(x-1)P(x-1))/P(x)$ and
$g(x)=\gcd(c(x),c(x-1))$, and set

$S=\sum_{n\geq 1}\dfrac{z^{n}}{c(n)P(n)P(n+1)}\;.$

Assume that $c(x)$, $P(x)$, and $P(x+1)$ are pairwise coprime. We have the
continued fraction $S=[[0,R(n)/g(n)],[r_{1},-zc(n)^{2}/(g(n)g(n+1))]]$ and the
identity $S=r_{2}+\sum_{n\geq 1}z^{n}d(n)/c(n)$ for some rational numbers
$r_{1}$ and $r_{2}$ and polynomial $d$ with $\deg(d)<\deg(c)$.

We also need the following conjecture:

###### Conjecture 4.2

Let $u$ be a nonnegative integers. Up to a multiplicative constant there
exists a unique polynomial $P$ such that

$\displaystyle(x+2u+4)(x+u+1)$ $\displaystyle P(x+1)-(x-1)(x+u-1)P(x-1)$
$\displaystyle=((2u^{2}+8u+7)x+(2u+3)(u^{2}+3u+1))P(x)\;,$

and we have $\deg(P)=u(u+3)$. The same is true if we require

$\displaystyle(x+2u+4)(x+u+4)$ $\displaystyle P(x+1)-(x-1)(x+u+2)P(x-1)$
$\displaystyle=((2u^{2}+8u+7)x+(2u+3)(u+2)(u+3))P(x)\;.$

Assuming this conjecture, in Proposition 4.1 we choose $z=-1$,
$c(x)=\prod_{0\leq i\leq 2u+4}(n+i)^{e_{i}}$ as before, with $m_{1}=u$,
$m_{2}=2$, and $m_{3}=u+3$, so $m=2u+5$, and $P$ the polynomial satisfying the
first identity above. We check that $g(x)=(x+u)\prod_{0\leq i\leq 2u+3}(x+i)$,
hence

$-zc(n)^{2}/(g(n)g(n+1))=n(n+u)(n+u+1)(n+2u+4)\;,$

and because of the identity for $P$,

$R(n)/g(n)=((2u^{2}+8u+7)n+(2u+3)(u^{2}+3u+1)\;.$

Finally, the identity for $S$ given by Proposition 4.1 implies that $S$ is a
linear combination of $\pi^{2}$, $\log(2)$, and $1$. Similarly for the second
identity. Thus, the other two one-parameter families follow from the above
conjecture together with the additional conjecture that $\log(2)$ does not
occur in the linear combination for $S$.

Remark: Concerning the above conjecture, Peter Mueller from the MathOverflow
forum proves the following:

  1. (1)

The two statements are equivalent.

  2. (2)

If $P$ exists it is indeed unique up to a multiplicative constant, and has
degree $u(u+3)$.

It is possible that there exist other, possibly more complicated, one-
parameter families.

##  5 CFs for $\pi^{2}$: Convergence in $1/((4/3)^{n}n^{P})$

Here we use the $\operatorname{asin}^{2}(z)$ series with $z=\sqrt{3}/2$,
giving the series

$\sum_{n\geq 1}\dfrac{3^{n}}{n^{2}\binom{2n}{n}}=\dfrac{2\pi^{2}}{9}\;,$

and applying Euler’s transformation shows that $2\pi^{2}/9$ is given by the CF
$((0,7n^{2}-8n+3),(3,-6n^{3}(2n-1)))$ with speed of convergence identical to
that of the series, in $3\sqrt{\pi}/((4/3)^{n}n^{3/2})$, and as before we can
write in abbreviated form

$\pi^{2}=(7n^{2}-8n+3,-6n^{3}(2n-1))\;.$

Applying Bauer–Muir does not give anything simple. So as before we search for
representations $\pi^{2}=(A(n),B(n))$ with $A(n)=7n^{2}+a_{1}n+a_{0}$ and
$B(n)=-6n(n+u_{1})(n+u_{2})(2n-1+2u_{3})$. Note that we could try other types
for $B(n)$ with the same leading term $-12n^{4}$ and completely factored, for
instance $B(n)=-n(2n+u_{1})(2n+u_{2})(3n+u_{3})$, but I have not found any new
representations in this way.

Here, the situation is less favorable since I have not been able to classify
completely the representations. I have found four two-parameter families:

$\displaystyle\pi^{2}$
$\displaystyle=(7n^{2}+(2u+3v-6)n-(5u^{2}+15vu+18v^{2}-2),$
$\displaystyle\phantom{=}-6n(n+u)(n+u+3v)(2n+1-4u-6v))$
$\displaystyle=(7n^{2}+(2u+3v-8)n-(5u^{2}+(15v+2)u+3(3v-1)(2v+1)),$
$\displaystyle\phantom{=}-6n(n+u)(n+u+3v)(2n-1-4u-6v))$
$\displaystyle=(7n^{2}+(15u+3v-8)n-3(2v+1)(3u+3v-1),$
$\displaystyle\phantom{=}-6n(n+u)(n+3u+3v)(2n-1-6v))$
$\displaystyle=(7n^{2}+(15u+3v-3)n-2(3v+4)(3u+3v+2),$
$\displaystyle\phantom{=}-6n(n+u)(n+3u+3v+3)(2n-5-6v))$

with all parameters nonnegative, plus a number of apparently sporadic cases:

$\displaystyle\pi^{2}$ $\displaystyle=(7n^{2}-16n+3,-6n^{3}(2n-5))$
$\displaystyle=(7n^{2}-19n-6,-6n^{2}(n+1)(2n-11))$
$\displaystyle=(7n^{2}+17n,-6n(n+3)^{2}(2n+1))$
$\displaystyle=(7n^{2}+14n-36,-6n(n+3)(n+5)(2n-7))$
$\displaystyle=(7n^{2}+54n+5,-6n(n+6)(n+10)(2n+1))$
$\displaystyle=(7n^{2}+31n-60,-6n(n+1)(n+10)(2n-3))\;,$

all with convergence speeds $C/((4/3)^{n}n^{P})$ for varying half-integral
$P$.

I have not tried to prove the validity of the above CFs.

The upshot of this case is that first, that I have probably not found all
possible cases, and second, that I have not seen the patterns leading either
to additional two-parameter families, or even possibly to a three-parameter
family.

Note, however, that in Section 7 we will meet the same _exact_ situation (four
two-parameter families plus six sporadic cases), which makes me believe that
there is a relation between the CFs of that section and those of the present
section.

##  6 CFs for $\pi^{2}$: Convergence in $1/(2^{n}n^{P})$

Here we use the $\operatorname{asin}^{2}(z)$ series with $z=\sqrt{2}/2$,
giving the series

$\sum_{n\geq 1}\dfrac{2^{n}}{n^{2}\binom{2n}{n}}=\dfrac{\pi^{2}}{8}\;,$

and applying Euler’s transformation shows that $\pi^{2}/8$ is given by the CF
$((0,3n^{2}-3n+1),(1,-n^{3}(2n-1)))$ with speed of convergence identical to
that of the series, in $\sqrt{\pi}/(2^{n}n^{3/2})$, and as before we can write
in abbreviated form

$\pi^{2}=(3n^{2}-3n+1,-n^{3}(2n-1))\;.$

Here, applying Bauer–Muir gives the one-parameter family
$\pi^{2}=(3n^{2}-3n+1,-n^{3}(2n-1-2k))$, and we search for representations
$\pi^{2}=(A(n),B(n))$ with $A(n)=3n^{2}+a_{1}n+a_{0}$ and
$B(n)=-n(n+u_{1})(n+u_{2})(2n-1+2u_{3})$. After some work, we find the three-
parameter family:

$\displaystyle\pi^{2}$ $\displaystyle=(3n^{2}+(8u+4v-3)n+(2u-1)(2u+2v-1),$
$\displaystyle\phantom{=}-n(n+2u)(n+2u+2v)(2n+2u-2k-1))\;,$

where all the parameters are nonnegative, with speed of convergence

$C/(2^{n}n^{u+2v+3/2+3k})$

for some constant $C$.

An extensive additional search seems to show that there are no additional
sporadic representations.

It is easy to give a _proof_ of the validity of the above family, which we
leave to the reader.

##  7 CFs for $\pi^{2}$: Convergence in $1/(4^{n}n^{P})$

Here we use the $\operatorname{asin}^{2}(z)$ series with $z=1/2$, giving the
series

$\sum_{n\geq 1}\dfrac{1}{n^{2}\binom{2n}{n}}=\dfrac{\pi^{2}}{18}\;,$

and applying Euler’s transformation shows that $\pi^{2}/18$ is given by the CF
$((0,5n^{2}-4n+1),(1,-2n^{3}(2n-1)))$ with speed of convergence identical to
that of the series, in $\sqrt{\pi}/3/(4^{n}n^{3/2})$, and as before we can
write in abbreviated form

$\pi^{2}=(5n^{2}-4n+1,-2n^{3}(2n-1))\;.$

Applying Bauer–Muir leads to an explicit complicated one-parameter family
which will be a special case of the ones below, and the situation is similar
to that of Section 5: I have found four two-parameter families:

$\displaystyle\pi^{2}$
$\displaystyle=(5n^{2}+(14u+21v-4)n+(3u+3v-1)(3u+6v-1),$
$\displaystyle\phantom{=}-2n(n+u)(n+u+3v)(2n-1))$
$\displaystyle=(5n^{2}+(14u+21v-6)n+9u^{2}+(27v-8)u+2(3v-1)^{2},$
$\displaystyle\phantom{=}-2n(n+u)(n+u+3v)(2n+1))$
$\displaystyle=(5n^{2}+(17u+21v+15)n+2(2u+3v+2)(3u+3v+2),$
$\displaystyle\phantom{=}-2n(n+u)(n+3u+3v+3)(2n+2u+1))$
$\displaystyle=(5n^{2}+(17u+21v-4)n+(3u+3v-1)(4u+6v-1),$
$\displaystyle\phantom{=}-2n(n+u)(n+3u+3v)(2n+2u-1))\;,$

with all parameters nonnegative, plus a number of apparently sporadic cases:

$\displaystyle\pi^{2}$ $\displaystyle=(5n^{2}+4n+1,-2n^{3}(2n+3))$
$\displaystyle=(5n^{2}+15n+6,-2n^{2}(n+1)(2n-1))$
$\displaystyle=(5n^{2}+19n+16,-2n(n+3)^{2}(2n+3))$
$\displaystyle=(5n^{2}+78n+261,-2n(n+3)(n+7)(2n-1))$
$\displaystyle=(5n^{2}+38n+68,-2n(n+3)(n+5)(2n+1))$
$\displaystyle=(5n^{2}+45n+60,-2n(n+1)(n+10)(2n+3))\;,$

all with convergence speeds $C/(4^{n}n^{P})$ for varying half-integral $P$.

As in Section 5, I have not tried to prove the validity of these CFs, I have
also probably not found all possible cases, and I have not seen the patterns
leading either to additional one- or two-parameter families, or even possibly
to a three-parameter family.

Nonetheless, the reader has certainly noticed that there is a striking
resemblance between the families of this section and those of Section 5, and
this should probably be investigated.

##  8 CFs for $\pi^{2}$: Convergence in $(-1)^{n}/(8^{n}n^{P})$

Let

$F_{n}=\sum_{0\leq k\leq n}\binom{n}{k}^{3}$

be the $n$th _Franel number_. From the recursion found by Franel for these
numbers and the computation of so-called Apéry limits (see for instance [2]),
we have the CF $\pi^{2}/3=((0,7n^{2}-7n+2),(8,8n^{4}))$ with speed of
convergence $C/(-8)^{n}$ and whose $n$th denominator $q(n)$ is given by
$q(n)=n!^{2}F_{n}$, giving the formula

$\dfrac{\pi^{2}}{3}=\sum_{n\geq
1}(-1)^{n-1}\dfrac{8^{n}}{n^{2}F_{n}F_{n-1}}\;.$

As usual we write the CF in abbreviated form

$\pi^{2}=(7n^{2}-7n+2,8n^{4})\;.$

Bauer–Muir acceleration gives something complicated, so as usual we search.
The situation is even worse here: I have found experimentally a large number
of representations, for which I was able to extract three two-parameter
families and a small number of one-parameter families, as follows:

$\displaystyle\pi^{2}$
$\displaystyle=(7n^{2}+((48-20v)u-7)n+2(4u-1)((8-5v)u-1),$
$\displaystyle\phantom{=((}8n(n+uv)(n+u(v+2))(n+4u))$
$\displaystyle=(7n^{2}+(24v-20u-7)n+(4v-2)(4v-5u-1),$
$\displaystyle\phantom{=((}8n(n+u)(n+2v)(n+u+v))$
$\displaystyle=(7n^{2}+(12u+20v-7)n+(4v-2)(3u+3v-1),$
$\displaystyle\phantom{=((}8n(n+2v)(n+3u+2v)(n+3u+3v))$
$\displaystyle=(7n^{2}+(30u-7)n+(3u-1)(9u-2),8n^{2}(n+3u)^{2})$
$\displaystyle=(7n^{2}+(34u-7)n+27u^{2}-17u+2,8n^{2}(n+u)^{2})$
$\displaystyle=(7n^{2}+(22u-7)n+(3u-1)(5u-2),8n(n+u)^{2}(n+3u))$
$\displaystyle=(7n^{2}+(58u-7)n+87u^{2}-29u+2,8n(n+u)^{2}(n+3u))$
$\displaystyle=(7n^{2}+(48u-23)n+2(4u+1)(8u-7),8n(n+2)(n+2u+3)(n+4u+2))$
$\displaystyle=(7n^{2}+(34u+17)n+3(3u+1)(3u+2),8n(n+1)(n+u)(n+u+2))\;,$

where $u$ and $v$ are nonnegative integers, except in the third two-parameter
representation where one can have $u\geq-\lfloor 2v/3\rfloor$, all with
convergence speeds $C/((-8)^{n}n^{P})$ for varying integral $P$.

I have found many (for now more than $60$) additional individual CFs, from
which one can probably extract more one-parameter families. Here is a small
sample:

$\displaystyle\pi^{2}$ $\displaystyle=(7n^{2}+26n+8,8n^{3}(n+3))$
$\displaystyle=(7n^{2}+80n+105,8n^{2}(n+1)(n+2))$
$\displaystyle=(7n^{2}+24n+9,8n^{2}(n+1)(n+4))$
$\displaystyle=(7n^{2}+65n+70,8n^{2}(n+1)(n+8))$
$\displaystyle=(7n^{2}-15n-12,8n^{2}(n+4)^{2})$
$\displaystyle=(7n^{2}+57n+96,8n^{2}(n+4)^{2})\;.$

##  9 CFs for $\pi^{2}$: Convergence in $1/(-((1+\sqrt{5})/2)^{10})^{n}$

This is the famous CF of Apéry:

$\pi^{2}=((0,11n^{2}-11n+3),(30,n^{4}))$

with convergence in $24\pi^{2}/(1+\sqrt{2})^{10n+5}$.

This CF seems to be unique of its kind, i.e. of the form
$\pi^{2}=(11n^{2}+a_{1}n+a_{0},n(n+u_{1})(n+u_{2})(n+u_{3}))$.

##  10 CFs for $\pi^{2}$: Other Speeds of Convergence

I have not found any convergent CF of the form $\pi^{2}=(A(n),B(n))$ with
$\deg(A)\leq 2$, $\deg(B)\leq 4$ and $A$ and $B$ with integral coefficients
other than the families mentioned above, up to trivial equivalence such as
changing $n$ into $n+j$ or multiplying $(A,B)$ by $(c,c^{2})$ for a nonzero
constant $c$. Of course, there exist infinitely many other CFs for $\pi^{2}$,
but of larger bidegrees.

##  11 CFs for $\zeta(3)$: Convergence in $1/n^{P}$

We now study CFs for $\zeta(3)$. Similarly to $\pi^{2}$, as far as the author
is aware, up to equivalence there exist exactly eight types of convergence of
polynomial continued fractions for $\zeta(3)$ with integral coefficients and
bidegree $\leq(3,6)$: three families with convergence in $1/n^{P}$ with
$P\in{\mathbb{Z}}$, a single CF with convergence in $(-1)^{n}/n^{P}$, a single
CF with convergence in $1/(4^{n}n^{P})$, a family of CFs with convergence in
$1/((-4)^{n}n^{P})$, a single CF with convergence in $1/E^{n}$ with
$(1+\sqrt{2})^{4}$, and finally the unique CF of Apéry with convergence in
$1/E^{n}$ for $E=(1+\sqrt{2})^{8}$. We consider each of these cases in turn.

###  11.1 First Family

The natural series giving $\zeta(3)$ is $\zeta(3)=\sum_{n\geq 1}1/n^{3}$, so
Euler’s transformation of series gives the CF
$\zeta(3)=((0,(2n-1)(n^{2}-n+1)),(1,-n^{6}))$, abbreviated as usual as

$\zeta(3)=((2n-1)(n^{2}-n+1),-n^{6})\;.$

Bauer–Muir acceleration leads to the family
$\zeta(3)=((2n-1)(n^{2}-n+2k^{2}+2k+1),-n^{6})$ for nonnegative integer $k$.

Instead of doing a search, we generalize the proof given in Section 2.1: using
the same notation, we choose $e_{i}=1$, $2$, $3$, $2$, or $1$ according to
$0\leq i<m_{1}$, $m_{1}\leq i<m_{1}+m_{2}$, $m_{1}+m_{2}\leq
i<m_{1}+m_{2}+m_{3}$, $m_{1}+m_{2}+m_{3}\leq i<m_{1}+m_{2}+m_{3}+m_{4}$, or
$m_{1}+m_{2}+m_{3}+m_{4}\leq i<m$ respectively, and this guarantees that the
CF $(A(n),B(n))$ will have bidegree $\leq(3,6)$, with

$\displaystyle A(n)$
$\displaystyle=(n+m_{1}+m_{2}+m_{3}-1)(n+m_{1}+m_{2}+m_{3}+m_{4}-1)(n+m-1)$
$\displaystyle\phantom{=}+(n-1)(n+m_{1}-1)(n+m_{1}+m_{2}-1)\text{\quad and}$
$\displaystyle B(n)$
$\displaystyle=-n(n+m_{1})(n+m_{1}+m_{2})(n+m_{1}+m_{2}+m_{3}-1)$
$\displaystyle\phantom{=}(n+m_{1}+m_{2}+m_{3}+m_{4}-1)(n+m-1)\;.$

However the sum $S$ will be a linear combination of $1$, $\pi^{2}$, and
$\zeta(3)$, and we do not want any $\pi^{2}$. For this, a _sufficient_
condition is that $m_{3}$ is odd and $m_{4}=m_{2}$ and $m_{5}=m_{1}$, giving
the parametric family $(A,B)$ with

$\displaystyle A(n)$ $\displaystyle=(n+u+v+2w)(n+u+2v+2w)(n+2u+2v+2w)$
$\displaystyle\phantom{=}+(n-1)(n+u-1)(n+u+v-1)\text{\quad and}$
$\displaystyle B(n)$
$\displaystyle=-n(n+u)(n+u+v)(n+u+v+2w)(n+u+2v+2w)(n+2u+2v+2w)\;,$

and note that

$\displaystyle A(n)$ $\displaystyle=(2n+2u+2v+2w-1)(n^{2}+(2u+2v+2w-1)n$
$\displaystyle\phantom{=}+(u^{2}+(3v+4w)u+2v^{2}+(6w+1)v+4w^{2}+2w+1))\;.$

Applying Bauer–Muir, we thus find the four-parameter family
$\zeta(3)=(A(n),B(n))$ with

$\displaystyle A(n)$
$\displaystyle=(2n+2(u+v+w)-1)(n^{2}+(2(u+v+w)-1)n+u^{2}+2v^{2}+4w^{2}$
$\displaystyle\phantom{=}+3uv+4uw+6vw+v+2w+1+2k(u+2v+3w)+2k^{2}+2k),$
$\displaystyle B(n)$
$\displaystyle=-n(n+u)(n+u+v)(n+u+v+2w)(n+u+2v+2w)(n+2u+2v+2w)\;,$

where $u$, $v$, $w$, and $k$ are nonnegative integer parameters, with speed of
convergence in $C/n^{2u+4v+6w+4k+2}$ for some constant $C$.

However, the sufficient condition (symmetry of
$(m_{1},m_{2},m_{3},m_{4},m_{5})$) above is not necessary: it is easy to find
as many nonsymmetrical examples as we like, the simplest being
$(m_{1},m_{2},m_{3},m_{4},m_{5})=(0,0,2,2,0)$. I have not yet been able to
give a reasonable classification of these additional examples, here is a small
sample:

$\displaystyle\zeta(3)$
$\displaystyle=(2n^{3}+4n^{2}+18n+8,-n^{3}(n+1)(n+3)^{2})$
$\displaystyle=(2n^{3}+8n^{2}+26n+28,-n^{2}(n+2)(n+3)^{3})$
$\displaystyle=(2n^{3}+11n^{2}+53n+60,-n^{2}(n+1)(n+2)(n+5)(n+6))$
$\displaystyle=(2n^{3}+17n^{2}+78n+119,-n(n+2)^{2}(n+3)(n+5)(n+8))$
$\displaystyle=(2n^{3}+19n^{2}+93n+180,-n(n+1)(n+4)(n+5)(n+6)^{2})$
$\displaystyle=(2n^{3}+20n^{2}+163n+299,-n^{3}(n+3)(n+10)^{2})$
$\displaystyle=(2n^{3}+22n^{2}+156n+360,-n(n+1)(n+2)(n+5)(n+8)(n+9))$
$\displaystyle=(2n^{3}+23n^{2}+201n+500,-n^{2}(n+1)(n+5)(n+10)^{2})\;.$

###  11.2 Second Family

We also have the simple modification obtained by removing the Euler factor at
$2$: $\zeta(3)=(8/7)\sum_{n\geq 1}1/(2n-1)^{3}$, so Euler’s transformation
gives the CF $\zeta(3)=((0,1,4(n-1)(4n^{2}-8n+7)),(8/7,-(2n-1)^{6}))$,
abbreviated as usual as

$\zeta(3)=(4(n-1)(4n^{2}-8n+7),-(2n-1)^{6})\;.$

Bauer–Muir acceleration leads to the family
$\zeta(3)=(4(n-1)(4n^{2}-8n+8k^{2}+8k+7),-(2n-1)^{6})$ for nonnegative integer
$k$. As in the case of $\pi^{2}$, to find parametric families we simply
replace $n$ by $n-1/2$ and multiply by suitable powers of $2$ in the first
family found above, giving the four-parameter family $\zeta(3)=(A(n),B(n))$
with

$\displaystyle A(n)$
$\displaystyle=4(n+u+v+w-1)(4n^{2}+8(u+v+w-1)n+4u^{2}+8v^{2}+16w^{2}$
$\displaystyle\phantom{=}+12uv+16uw+24vw-4u+4w+8k(u+2v+3w)+8k^{2}+8k+7),$
$\displaystyle B(n)$
$\displaystyle=-(2n-1)(2n+u-1)(2n+u+v-1)(2n+u+v+2w-1)\cdot$
$\displaystyle\phantom{=-}\cdot(2n+u+2v+2w-1)(2n+2u+2v+2w-1)\;,$

where $u$, $v$, $w$, and $k$ are nonnegative integer parameters, with speed of
convergence in $C/n^{2u+4v+6w+4k+2}$ for some constant $C$, and of course as
in the first family, one also has the numerous additional apparently sporadic
cases, such as

$\zeta(3)=(16n^{3}+8n^{2}+124n-2,-(2n-1)^{3}(2n+1)(2n+5)^{2})\;.$

###  11.3 Third Family

From work of Y. Yang [4], one finds the CF

$\zeta(3)=((0,(2n-1)(2n^{2}-2n+1)),(2/7,-4n^{6}))$

with extremely slow convergence in $C/\log(n)$, which as usual we abbreviate
to $\zeta(3)=((2n-1)(2n^{2}-2n+1),-4n^{6})$. Modifying his construction and
applying Bauer–Muir, we find the two-parameter family $\zeta(3)=(A(n),B(n))$
with

$\displaystyle A(n)$
$\displaystyle=(2n+2u-1)(2n^{2}+(4u-2)n+(2k+2)u+k^{2}+4k+5),$ $\displaystyle
B(n)$ $\displaystyle=-4n^{3}(n+2u)^{3}\;,$

with $u$ and $k$ nonnegative, with speed of convergence in $C/n^{2u+2k+4}$.

It is interesting to compare this with the very similar two-parameter family
$\zeta(3)=(A(n),B(n))$ obtained by setting $u=v=0$ and replacing $w$ by $u$ in
Section 11.1:

$\displaystyle A(n)$
$\displaystyle=(2n+2u-1)(2n^{2}+(4u-2)n+8u^{2}+(12k+4)u+4k^{2}+4k+2),$
$\displaystyle B(n)$ $\displaystyle=-4n^{3}(n+2u)^{3}\;,$

with $u$ and $k$ nonnegative, with speed of convergence in $C/n^{6u+4k+2}$.

##  12 CFs for $\zeta(3)$: Convergence in $(-1)^{n}/n^{P}$

Using Euler’s transformation of series into CFs but now applied to
$(3/4)\zeta(3)=\sum_{n\geq 1}(-1)^{n-1}/n^{3}$, we see that $\zeta(3)$ is
given by the CF $((0,3n^{2}-3n+1),(4/3,n^{6}))$, with speed of convergence
identical to that of the series, in $(-1)^{n}2/(3n^{3})$, which we write in
abbreviated form

$\zeta(3)=(3n^{2}-3n+1,n^{6})\;.$

There is apparently no other CF of this type with convergence in
$(-1)^{n}/n^{P}$. The main reason is probably that, if we try to apply similar
constructions to those that we used for $\sum_{n\geq 1}(-1)^{n-1}/n^{2}$, we
are now going to obtain linear combinations of $1$, $\zeta(3)$, and the two
additional “parasitic” terms $\zeta(2)$ and $\log(2)$, instead of simply
$\log(2)$ alone.

##  13 CFs for $\zeta(3)$: Convergence in $1/(4^{n}n^{P})$

Let

$D_{n}=\sum_{0\leq k\leq n}\binom{n}{k}^{2}\binom{2k}{k}\binom{2n-2k}{n-k}$

be the $n$th so-called _Domb number_. From the recursion that one can find for
these numbers and the computation of Apéry limits (see for instance [2]), we
have the CF $\zeta(3)=((0,(2n-1)(5n^{2}-5n+2)),(12/7,-16n^{6}))$, whose $n$th
denominator $q(n)$ is given by $q(n)=n!^{3}D_{n}/2^{n}$, giving the formula

$\dfrac{56\zeta(3)}{3}=\sum_{n\geq 1}\dfrac{64^{n}}{n^{3}D_{n}D_{n-1}}\;.$

As usual we write the CF in abbreviated form

$\zeta(3)=((2n-1)(5n^{2}-5n+2),-16n^{6})\;.$

Bauer–Muir acceleration gives something complicated, and as above I have not
found any other CF of this type with convergence in $1/(4^{n}n^{P})$.

##  14 CFs for $\zeta(3)$: Convergence in $(-1)^{n}/(4^{n}n^{P})$

From the famous series mentioned by Apéry but known before him

$\zeta(3)=\dfrac{5}{2}\sum_{n\geq 1}\dfrac{(-1)^{n-1}}{n^{3}\binom{2n}{n}}\;,$

one deduces the CF $\zeta(3)=((0,3n^{3}+n^{2}-3n+1),(5/2,2n^{5}(2n-1)))$ with
the same speed of convergence as the series in
$\sqrt{\pi}/2/((-4)^{n}n^{5/2})$, which we write as

$\zeta(3)=(3n^{3}+n^{2}-3n+1,2n^{5}(2n-1))\;.$

Here, a variant of Bauer–Muir gives the family

$\zeta(3)=(3n^{3}+(11k+1)n^{2}-(11k+3)n+3k+1,n^{4}(2n-k)(2n-k-1))$

with convergence in $C/((-4)^{n}n^{5k+5/2})$. A search apparently does not
give any other CF.

##  15 CFs for $\zeta(3)$: Convergence in $1/(1+\sqrt{2})^{4n}$

Let

$C_{n}=\sum_{n/2\leq k\leq n}\binom{n}{k}^{2}\binom{2k}{n}^{2}$

(these numbers do not seem to have a name). From the recursion that one can
find for these numbers and the computation of Apéry limits (see for instance
[2]), we have the CF $\zeta(3)=((0,(2n-1)(3n^{2}-3n+1)),(8/7,-n^{6}))$ with
speed of convergence $4\pi^{3}/7/(1+\sqrt{2})^{4n+2}$, whose $n$th denominator
$q(n)$ is given by $q(n)=n!^{3}C_{n}/4^{n}$, giving the formula

$\dfrac{7\zeta(3)}{2}=\sum_{n\geq 1}\dfrac{16^{n}}{n^{3}C_{n}C_{n-1}}\;.$

As usual we write the CF in abbreviated form

$\zeta(3)=((2n-1)(3n^{2}-3n+1),-n^{6})\;.$

Since the exponent of the speed of convergence is irrational, Bauer–Muir
acceleration would give irrational coefficients so is not applicable, and I
have not been able to find other CFs of this type.

##  16 CFs for $\zeta(3)$: Convergence in $1/(1+\sqrt{2})^{8n}$

This is the famous CF of Apéry:

$\zeta(3)=((0,(2n-1)(17n^{2}-17n+5)),(6,-n^{6}))$

with convergence in $4\pi^{3}/(1+\sqrt{2})^{8n+4}$.

This CF seems to be unique of its kind, i.e. of the form
$\zeta(3)=(34n^{3}+a_{2}n^{2}+a_{1}n+a_{0},-n(n+u_{1})(n+u_{2})(n+u_{3})(n+u_{4})(n+u_{5})(n+u_{6}))$.

##  17 CFs for $\zeta(3)$: Other Speeds of Convergence

I have not found any convergent CF of the form $\zeta(3)=(A(n),B(n))$ with
$\deg(A)\leq 3$, $\deg(B)\leq 6$ and $A$ and $B$ with integral coefficients
other than the families mentioned above, up to trivial equivalence such as
changing $n$ into $n+j$ or multiplying $(A,B)$ by $(c,c^{2})$ for a nonzero
constant $c$. Of course, there exist infinitely many other CFs for $\zeta(3)$,
but of larger bidegrees.

##  18 Other Periods

All the preceding sections dealt with CFs for $\pi^{2}$ and $\zeta(3)$. A
similar study can be done for other periods. I briefly summarize what I have
found, restricting as usual to CFs of bidegrees $\leq(k,2k)$ for periods of
degree $k$.

###  18.1 Periods of Degree $1$

As just mentioned, we restrict to bidegrees $\leq(1,2)$.

$\bullet$ $z=\pi$: one finds parametric families of CFs with convergence in
$(-1)^{n}/n^{P}$, $1/(2^{n}n^{P})$, and $(-1)^{n}/(1+\sqrt{2})^{2n}$.

$\bullet$ $z=\pi/\sqrt{3}$: one finds parametric families of CFs with
convergence in $1/((4/3)^{n}n^{P})$, $(-1)^{n}/(3^{n}n^{P})$,
$1/(4^{n}n^{P})$, and $(-1)^{n}/(2+\sqrt{3})^{2n}$.

$\bullet$ $z=\log(2)$: one finds parametric families of CFs with convergence
in $(-1)^{n}/n^{P}$, $1/(2^{n}n^{P})$, $(-1)^{n}/(8^{n}n^{P})$,
$1/(9^{n}n^{P})$, and $1/(1+\sqrt{2})^{4n}$.

Note that one can also study CFs for $z=\log(m)$ for other rational values of
$m$.

$\bullet$ $z=\sqrt{2}\log(1+\sqrt{2})$: one finds parametric families of CFs
with convergence in $1/(1+\sqrt{2})^{2n}$.

###  18.2 Periods of Degree $2$

We restrict to bidegrees $\leq(2,4)$.

$\bullet$ $z=\pi^{2}$: see the extensive lists given in the preceding
sections.

$\bullet$ $z=L(\chi_{-3},2)$: one finds parametric families of CFs with
convergence $1/((9/8)^{n}n^{P})$ and $1/(9^{n}n^{P})$.

$\bullet$ $z=L(\chi_{-4},2)=G$, Catalan’s constant: one finds parametric
families of CFs with convergence $1/n^{P}$, $(-1)^{n}/n^{P}$, and
$1/(2^{n}n^{P})$.

###  18.3 Periods of Degree $3$

We restrict to bidegrees $\leq(3,6)$.

$\bullet$ $z=\zeta(3)$: see the extensive lists given in the preceding
sections.

$\bullet$ For other periods of degree $3$ such as $\pi^{3}$ and
$\pi^{3}/\sqrt{3}$, either the CFs are isolated, or are one-parameter families
coming from Bauer–Muir acceleration.

I would evidently welcome any additional data and comments on the subject of
this paper.

## References

  * [1] [Coh] H. Cohen, Elementary continued fractions for linear combinations of zeta and $L$ values, arXiv:2212.01095 (2022).
  * [2] [Gor] O. Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 (2021).
  * [3] [Kon-Zag] M. Kontsevitch and D. Zagier, Periods, in Mathematics unlimited 2001 and Beyond, Springer (2000), 771–808.
  * [4] [Yan] Y. Yang, Apéry limits and special values of $L$-functions, J. Math. Anal. Appl. 343 (2008), 492–513.