\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 139, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/139\hfil Eigenvalue problems]
{Eigenvalue problems with $p$-Laplacian operators}

\author[Y.-H. Cheng \hfil EJDE-2014/139\hfilneg]
{Yan-Hsiou Cheng}  % in alphabetical order

\address{Yan-Hsiou Cheng\\
 Department of Mathematics and Information Education\\
 National Taipei University of Education\\
 Taipei City 106, Taiwan}
\email{yhcheng@tea.ntue.edu.tw}

\thanks{Submitted November 7, 2013. Published June 16, 2014.}
\subjclass[2000]{34A55, 34L15}
\keywords{$p$-Laplacian; inverse spectral problem; instability interval}

\begin{abstract}
 In this article, we study eigenvalue problems with the  $p$-Laplacian operator:
 $$
 -(|y'|^{p-2}y')'= (p-1)(\lambda\rho(x)-q(x))|y|^{p-2}y
 \quad \text{on } (0,\pi_{p}),
 $$
 where   $p>1$ and $\pi_{p}\equiv  2\pi/(p\sin(\pi/p))$.
 We show that if  $\rho \equiv 1$ and $q$ is single-well with transition
 point $a=\pi_{p}/2$, then the second Neumann eigenvalue is greater
 than or equal to the first Dirichlet eigenvalue; the equality holds
 if and only if $q$ is constant.
 The same result also holds for $p$-Laplacian problem with
 single-barrier   $\rho$ and $q \equiv 0$. Applying these results,
 we extend and improve a result by \cite{S07}
 by using finitely many  eigenvalues and by generalizing the string
 equation to $p$-Laplacian problem. Moreover, our results also extend
 a result of Huang \cite{H97}  on the estimate of the first instability
 interval for Hill equation to single-well function $q$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

Recently there are many studies on the $p$-Laplacian operator:
\begin{equation}-(|y'|^{p-2}y')'= (p-1)(\lambda\rho(x)
-q(x))|y|^{p-2}y\quad \text{ on } (0,\pi_{p}),
\label{eq1.1}
\end{equation}
where   $p>1$ and $\pi_{p}\equiv  2\pi/(p\sin(\pi/p))$.
An application for \eqref{eq1.1}, the most cited nowadays, is that of a
highly viscid fluid flow (cf. Ladyzhenskaya
\cite{la}, and Lions \cite{li}).
This involves partial differential equations, but for symmetric
flows, only the ordinary differential operator (perhaps in radial
form) is involved (see, e.g., Binding and Dr\'{a}bek \cite{BD2003},
del Pino, Elgueta and Manasevich \cite{dem}, del Pino and Manasevich
\cite{dm}, Rabinowitz \cite{r}, and Walter \cite{W98}).


In  1979, Elbert \cite{E79} showed that the inverse function 
$S_{p} (x)\equiv w$ of the  integral
$$
x=\int_0^{w}\frac{dt}{(1-t^{p})^{1/p}}\qquad
\text{for}\ 0\leq w\leq 1\,,$$
satisfies the initial valued problem
\begin{equation*}
\label{eq3.1}
  -(|u'|^{p-2}u')' = (p-1)|u|^{p-2}u ,\qquad u(0)=0 ,\ u'(0)=1.
\end{equation*}
The function $S_{p} (x)$ is called a generalized sine function
 and the value $\pi_{p}  \equiv 2
\int_0^{1} (1-t^{p})^{-1/p} dt = 2\pi/(p
\sin(\pi/p))$ is the first zero of $S_{p}(x)$.
 Continuing $S_{p}(x)$ symmetrically over
$x\in [ \pi_{p}/2,\pi_{p}] $ and antisymmetrically outside $[0,\pi_{p}]$ 
by defining
$$
S_{p}(x)=\begin{cases}
S_{p}(\pi_{p}-x),  & \text{if } \frac{\pi_{p}}{2}\leq x\leq \pi_{p}\,,\\
-S_{p}(x-\pi_{p}), & \text{if } \pi_{p}\leq x\leq 2\pi_{p}\,,
 \end{cases}
$$
and $S_{p}(x)=S_{p}(x-2n\pi_{p})$ for $n=\pm 1, \pm 2, \dots$, he
obtained a sine-like function defined on $\mathbb{R}$.
Furthermore, he found the Pythagorean trigonometric identity for
$p$-version:
\begin{equation*}
\label{eq2.3}
|S_{p} (x)|^{p}+|S_{p}' (x)|^{p}=1\,.
\end{equation*}

Similarly, it may be defined an analogue of the hyperbolic sine
function (see \cite{lind}) $Sh_{p} (x)\equiv v$ by the inverse
function of the integral $x=\int_0^{v} (1+|t|^{p})^{-1/p} dt$.
 It is clearly that $Sh_{p} (x)=(-1)^{-1/p}S_{p} ((-1)^{1/p}x)$ and 
$Sh_{p}' (x)=S_{p}' ((-1)^{1/p}x)$ where $(-1)^{1/p}=e^{\pi i/p}$.
 Furthermore, we have
 $Sh_{p}''(x)=\frac{|Sh_{p}(x)|^{p-2}Sh_{p}(x)}{Sh_{p}'^{p-2}(x)}$
  and
\[ %\label{eq2.5}
Sh_{p}'^{p}(x)-|Sh_{p}(x)|^{p}=1.
\]

Denote by $\sigma_{2k}$ ($\sigma_{2k-1}$) the set of periodic
(anti-periodic) eigenvalues of \eqref{eq1.1} which admit the
corresponding eigenfunctions with exactly $2k$ zeros in
$[0,\pi_{p})$. In 2001, Zhang \cite{Z01}  used a rotation number
function to show the existence of the minimal eigenvalue
$\underline{\lambda}_{n}=\min \sigma_{n}$ and the maximal
eigenvalue $\overline{\lambda}_{n}=\max \sigma_{n}$, respectively.
In particular, Binding and Rynne in a series of papers
\cite{BR07,BR08,BR09} showed that \eqref{eq1.1} has an infinite sequence of
variational and non-variational periodic eigenvalues and the
multiplicity of the periodic eigenvalue can be arbitrary. They
also showed that the Dirichlet eigenvalues $\{\mu_{n}\}_{n\geq
1}$ and  Neumann eigenvalues $\{\nu_{n}\}_{n\geq 0}$  for
\eqref{eq1.1}  acting on $(0,\pi_{p})$ satisfy 
\begin{gather*}
\dots \leq \overline{\lambda}_{2n-2}<
\underline{\lambda}_{2n-1} \leq   \mu_{2n-1},\\
\nu_{2n-1} \leq \overline{\lambda}_{2n-1}
<\underline{\lambda}_{2n} \leq   \mu_{2n},
\nu_{2n} \leq \overline{\lambda}_{2n}<\underline{\lambda}_{2n+1}\leq \dots.
\end{gather*}
 Note that, for $q\equiv 0$ and $\rho \equiv 1$, we find 
$\nu_0=0$ and  $\mu_{n}=\nu_{n}=n^{p}$ for $n\geq 1$.



Recently, the   eigenvalue gap/ratio are concerned. We say a
function $f$ is single-well with transition point $a$
if $f$ is decreasing on $(0,a)$ and increasing on $(a,\pi_{p})$;
$f$ is single-barrier if $-f$ is single-well. In 2010, Bogn\'ar
and Dosly  \cite{BD10} used the Pr\"ufer transformation derived
by generalized sine function to show that the Dirichlet
eigenvalues  for
\eqref{eq1.1} with $\rho \equiv 1$ and nonnegative single-well  
$q(x)$ satisfy $\mu_{n}/\mu_{m} \leq n^p/m^p$. 
Furthermore, Chen, Law, Lian and Wang
\cite{CLLW13} also used  the generalized  Pr\"ufer
transformation to show that $\mu_{n}/\mu_1\leq n^p$  for
\eqref{eq1.1} with $\rho \equiv 1$ and nonnegative continuous $q(x)$.
 On the other hand, the authors  in \cite{CLW} studied
 the first two Dirichlet eigenvalues  for
\eqref{eq1.1} and showed that 
(i) $\mu_2-\mu_1 \geq 2^{p}-1$ if $\rho \equiv 1$ and $q(x)$
 is single-well with transition point at $\pi_{p}/2$; 
(ii) $\mu_2/\mu_1 \geq 2^{p}$ if $q(x)
\equiv 0$ and $\rho(x)$ is single-barrier  with
transition point at $\pi_{p}/2$.


In this article, we study the gap between the Dirichlet eigenvalues
and Neumann eigenvalues. In
\cite[Theorem 2.5]{S07}, Shen considered the spectra
$\sigma_{D} = \{ \mu_1,\mu_2, \dots \}$,
$\sigma_{DN} =\{ \tau_1,\tau_2, \dots \}$,
$\sigma_{ND}= \{ \gamma_1,\gamma_2, \dots \}$, and
$\sigma_{N} =\{\nu_0, \nu_1,\nu_2, \dots \}$
for  the following string equations
\begin{gather*} %\label{eq1.5}
y''(x) +\mu \rho(x)y(x)=0,\quad y(0)=y(\pi)=0,\\
%\label{eq1.6}
u''(x) +\tau \rho(x)u(x)=0,\quad u(0)=u'(\pi)=0,\\
%\label{eq1.7}
z''(x) +\gamma \rho(x)z(x)=0,\quad z'(0)=z(\pi)=0,\\
%\label{eq1.8}
v''(x) +\nu \rho(x)v(x)=0,\quad v'(0)=v'(\pi)=0,
\end{gather*}
respectively, where $\rho$ is a positive piecewisely continuous
function defined on $[0, \pi]$. He showed that if 
$\sigma_{DN}= \sigma_{ND}$ and 
$\sigma_{N}= \sigma_{D}\cup \{0\}$, then $\rho (x)$ is a
constant function at its points of continuity.

Consider  \eqref{eq1.1} and assume $q$ and $\rho$ satisfy 
(i) $\rho\equiv 1$ and $q$ is single-well with transition point
$a=\pi_{p}/2$, or (ii) $q\equiv 0$ and $\rho$ is single-barrier
with transition point $a=\pi_{p}/2$.  In this paper, we show that
$\mu_1=\nu_1$ if and only if (i) $q$ is constant, or
(ii) $\rho$ is constant, respectively.  Our results extend and improve
the result of Shen \cite[Theorem 2.5]{S07} by using finitely many
eigenvalues and by generalizing the string equation to
$p$-Laplacian eigenvalue problem.


\begin{theorem} \label{thm1.1} 
Consider \eqref{eq1.1} with $q(x)\in L^{1}(0,\pi_{p})$ and $\rho\equiv 1$.
If  $q(x)$ is single-well on $(0,\pi_{p})$ with transition point $a=\pi_{p}/2$,
then $\nu_1\geq \mu_1$. Equality holds if and only if $q$
is constant. If $a\neq \pi_{p}/2$, then there exist some functions
$q$ giving $\nu_1< \mu_1$.
\end{theorem}


\begin{theorem} \label{thm1.2}
Consider   \eqref{eq1.1} with a positive piecewisely continuous
function $\rho$ and $q\equiv 0$. If
 $\rho(x)$ is single-barrier on $(0,\pi_{p})$ with transition point
$a=\pi_{p}/2$, then $\nu_1\geq \mu_1$. Equality holds if
and only if $\rho$ is  constant. If $a\neq \pi_{p}/2$, then there exist some 
functions $\rho$ giving $\nu_1< \mu_1$.
\end{theorem}

The proof of Theorem \ref{thm1.1}  follows the method developed by
 Horv\'ath \cite{H02}. We first perturb the extremal function $q$ 
and study the identity for $\frac{d}{dt}(\nu_1(t)-\mu_1(t))$ where
$t$ is a parameter. We will show that  the optimal function $q$  
is a step function with a jump at $\pi_{p}/2$ and then compel it to be constant.
Furthermore, by the principle of duality, the same method also
works for  \eqref{eq1.1} with $q\equiv 0$ and single-barrier
$\rho$.


We shall remark that Theorems \ref{thm1.1} and \ref{thm1.2}  can be
used to solve inverse problems of the instability interval for
$p=2$:
\begin{equation}
-y''=(\lambda\rho(x) -q(x))y\,.\label{eq1.10}
\end{equation}
Denote by  $\{\lambda_{n}\}_{n\geq 0}$ and
$\{\lambda'_{n}\}_{n\geq 1}$ the eigenvalues of \eqref{eq1.10}
with $q(x)=q(x+\pi), \rho(x)=\rho(x+\pi)$ under the periodic ($y(0)=y(\pi)$,
$y'(0)=y'(\pi)$), and anti-periodic ($y(0)=-y(\pi)$,
$y'(0)=-y'(\pi)$) boundary conditions, respectively.
 It is known \cite{CL55}  (see also \cite{B46,MW66}) that  
 $\nu_0\leq \lambda_0$ and
\begin{equation}
\begin{gathered}
\dots\leq\lambda_{2n-2} <\lambda'_{2n-1} \leq \nu_{2n-1}, \\
\mu_{2n-1} \leq \lambda'_{2n}< \lambda_{2n-1} \leq \nu_{2n},  \\
\mu_{2n} \leq \lambda_{2n}<\lambda'_{2n+1}\leq
\dots\\
\end{gathered}\label{eq1.12}
\end{equation}
The intervals $(\lambda'_{2n-1},\lambda'_{2n})$ and
$(\lambda_{2n-1},\lambda_{2n})$ are called the $(2n-1)$-th and
$2n$-th instability intervals. The interval
$(-\infty,\lambda_0)$ is called the zero-th instability
interval.

In 1946, Borg \cite{B46} studied an inverse problem for Hill's
equation. He showed that the potential $q(x)$ is constant if and
only if all instability intervals, except the zero-th, are
absent. Later, Hochstadt \cite{H84} generalized Borg's result and
showed that if $q$ is $C^{1}$, then $q$ has period $1/n$ if and
only if all those finite instability intervals whose index is not
a multiple of $n$ vanish.   In 1997, Huang \cite{H97} proved that
if $q$ is   symmetric single-well (or symmetric single-barrier),
then $q$ is constant if and only if the first instability
interval is absent, i.e. $\lambda'_1=\lambda'_2$. Thus, for
all instability intervals, the first instability gives the most
information about the potential $q$. Using Theorems
\ref{thm1.1} and \ref{thm1.2},  and \eqref{eq1.12}, we may eliminate the
assumption on the symmetric of $q$ and obtain the following
results immediately.

\begin{corollary} \label{coro1.3}
 Consider \eqref{eq1.10} with  $\pi$-periodic
functions $\rho$ and $q$. Then the first instability interval is
absent if and only if one of the following conditions holds:
\begin{itemize}
  \item[(i)] $\rho\equiv 1$ and $q$ is single-well with transition point $a=\pi/2$.

  \item[(ii)] $q\equiv 0$ and $\rho$ is single-barrier  with transition point
$a= \pi/2$.
\end{itemize}
\end{corollary}

The paper is organized as follows. In section 2, we use a
modified Pr\"ufer substitution and comparison theorem to derive
 properties of eigenfunctions. In section 3, we  study two
generalized trigonometric  equations. The Dirichlet and Neumann
eigenvalues are corresponding to the roots of  two generalized
trigonometric equations, respectively. Finally, in section 4, we
give  proofs of our main theorems
\ref{thm1.1} and \ref{thm1.2}.

\section{Preliminaries}

At the beginning of this section, we give two formulas of
generalized trigonometric functions. The proof is similar to the
classical trigonometric functions, so we omit it here.


\begin{lemma} \label{lem2.1}
Define the generalized tangent function by $T_{p}(x)\equiv
S_{p}(x)/ S_{p}'(x)$ for $x\neq (k+1/2)\pi_{p}$ and the
generalized reciprocal tangent function by
$RT_{p}(x)\equiv S_{p}'(x)/ S_{p}(x)$ for $x\neq k\pi_{p}$. Then we have
\begin{itemize}
  \item[(i)] $T_{p}'(x)=1+|T_{p}(x)|^{p}  .$

  \item[(ii)] $RT_{p}'(x)= -(RT_{p}(x))^{2} (1+|T_{p}(x)|^{p}).$
\end{itemize}
\end{lemma}

Denote by $(\mu_i, \phi_i)_{i\geq 1}$ the normalized Dirichlet
eigenpair and $(\nu_i, \psi_i)_{i\geq 0}$ the normalized
Neumann eigenpair of \eqref{eq1.1} with $\phi_i(x)>0$,
$\psi_i(x)>0$ for $x$ near $0^{+}$. The normalized condition means
$\int_0^{\pi_{p}}\rho(x)|\phi_i(x)|^{p}dx
=\int_0^{\pi_{p}}\rho(x)|\psi_i(x)|^{p}dx =1 $  for all $i$.




\begin{definition} {def1} \rm
Let $f$ and $g$ be continuous functions and $g(x)\neq 0$. Define
$h(x)\equiv f(x)/g(x)$.  We say $\alpha_0$ is a crossing point
of $f$ and $g$ if $h(\alpha_0)=1$ and $h$ satisfies one of the
following conditions
\begin{itemize}
  \item[(i)]
 $h(\alpha_0^{+})>1$ and  $h(\alpha_0^{-})<1$.
  \item[(ii)]  $h(\alpha_0^{+})<1$ and  $h(\alpha_0^{-})>1$.
\end{itemize}
\end{definition}

\begin{lemma} \label{lem2.2}
There are exactly two crossing points of $|\phi_1(x)|$ and
$|\psi_1(x)|$ in $(0,\pi_{p})$.
\end{lemma}

\begin{proof}
First,  we introduce a generalized Pr\"ufer substitution derived by $S_{p}$ and
 $S_{p}'$:
\begin{gather*}
 \phi_1(x)=r(x) S_{p}(\theta_{D}(x))\,,\quad \phi_1'(x) =
 r(x)S_{p}'(\theta_{D}(x))\,, \\
\psi_1(x)=R(x) S_{p}(\theta_{N}(x))\,,\quad \psi_1'(x) =
 R(x)S_{p}'(\theta_{N}(x))\,,
\end{gather*}
where $\theta_{D}(0)=0$ and $\theta_{N}(0)=\pi_{p}/2$. Here,
$\theta_{D}(x)$ and $\theta_{N}(x)$ are called the Pr\"ufer angles
of $\phi_1(x)$ and  $\psi_1(x)$, respectively. By direct
calculation, we find that
\begin{gather}
\theta_{D}'(x)=|S_{p}'(\theta_{D}(x))|^{p}
 +(\mu_1\rho(x)-q(x))|S_{p}(\theta_{D}(x))|^{p},\label{eq2.0.1}\\
\theta_{N}'(x)=|S_{p}'(\theta_{N}(x))|^{p}
 +(\nu_1\rho(x)-q(x))|S_{p}(\theta_{N}(x))|^{p}.\label{eq2.0.2}
\end{gather}

Let $x_0$ be the unique zero of $\psi_1(x)$ in $(0,\pi_{p})$.
Since $\phi_1(x)>0$ on $(0,x_0)$, $0=\phi_1(0)<\psi_1(0)$
and $\phi_1(x_0)>\psi_1(x_0)=0$, we find the number of the
crossing points of $\phi_1(x)$ and $\psi_1(x)$ in $(0,x_0)$
must be odd. Assume $0<x_1<x_2<x_{3}<x_0$ are   crossing
points of $\phi_1(x)$ and $\psi_1(x)$. Define
 $v(x)\equiv\frac{\psi_1(x)}{\phi_1(x)}$. Then $v(x_i)=1$ for
$i=1,2,3$. By Rolle's Theorem, there are 
$z_i\in (x_i,x_{i+1})$, $i=1,2$, such that $v'(z_i)=0$. Note that 
$$
v'(x)=\frac{\psi_1(x)\phi_1(x)}{\psi_1^{2}(x)}
[\frac{\phi_1'(x)}{\phi_1(x)}-\frac{\psi_1'(x)}{\psi_1(x)}]
=\frac{\phi_1(x)}{\psi_1(x)}[RT_{p}(\theta_{D}(x))-RT_{p}(\theta_{N}(x))].
$$
 Hence, we find $\theta_{D}(z_i)=\theta_{N}(z_i)$ for $i=1,2$. By
 applying  Comparison theorem
\cite{br89} on \eqref{eq2.0.1} and  \eqref{eq2.0.2}, we obtain 
$\mu_1=\nu_1$. This implies that
$\theta_{D}(x)=\theta_{N}(x)$ for all $x\in (0,x_0)$. But this
is a contradiction to $\theta_{D}(0)=0$ and
$\theta_{N}(0)=\pi_{p}/2$. Hence there is exactly one crossing
point of $\phi_1(x)$ and $\psi_1(x)$ in $(0,x_0)$.

Similarly, there is also exactly one crossing point of
$|\phi_1(x)|$ and $|\psi_1(x)|$ in $(x_0, \pi_{p})$. The
proof is complete.
\end{proof}

According to Lemma \ref{lem2.1}, we denote the points
$0<x_{-}<x_0<x_{+}<\pi_{p}$ such that $\psi_1(x_0)=0$ and
\begin{equation}
\label{eq2.1}
|\psi_1(x)|-|\phi_1(x)| \{\begin{cases}
\geq 0 &\text{on } (0,x_{-})\cup (x_{+},\pi_{p}),\\
\leq 0 &\text{on } (x_{-}, x_{+}). 
\end{cases}
\end{equation}

The following lemma is a $p$-version formula while the similar
formulas were derived in \cite{L94} and \cite{K76} for the
Schr\"odinger equation and string equation, respectively. The
argument is similar so we omit here.

\begin{lemma} \label{lem2.3}
Consider  \eqref{eq1.1} coupled with Dirichlet  or Neumann
boundary conditions
 on $(0,\pi_{p})$. Let $q(\cdot ,t)$ be a one-parameter family of
continuous functions and $\rho(\cdot , t)$ be a one-parameter
family of continuous functions such that $\frac{\partial
q}{\partial t}(x,t)$ and $\frac{\partial
\rho}{\partial t}(x,t)$ exist. Then
\begin{equation} \label{eq2.2}
\frac{d}{dt}\lambda(t)
=-\lambda (t) \int_0^{\pi_{p}}\frac{\partial \rho}{\partial t}(x,t)|y(x,t)|^{p}dx
+ \int_0^{\pi_{p}}\frac{\partial q}{\partial t}(x,t)|y(x,t)|^{p}dx\,.
\end{equation}
\end{lemma}

The following lemma will be used in the proofs of Theorems
\ref{thm1.1} and  \ref{thm1.2}. This lemma makes those proofs
simpler.

\begin{lemma}\label{lem2.4}
  Consider \eqref{eq1.1}. If    $q(x)$ is increasing and
   $\rho (x)$  satisfies $\rho (x)\geq \rho
(\pi_{p}-x)$ for $x\in (0,\pi_{p}/2)$, then $x_0\leq
\pi_{p}/2$.
\end{lemma}

\begin{proof}
Denote $Q_1(x)\equiv (p-1)(\nu_1\rho(x)-q(x)),\ Q_2(x)\equiv
Q_1(\pi_{p}-x),\ z_1(x)\equiv\psi_1(x)$, and
$z_2(x)\equiv -\psi_1(\pi_{p}-x)$. Then $Q_2(x)\leq
Q_1(x)$ on $(0,\min\{x_0,\pi_{p}-x_0\}]$ and we have the
following two problems
\begin{gather*}
(|z_1'|^{p-2}z_1')'+Q_1(x)|z_1|^{p-2}z_1=0
\quad\text{on }[0,x_0]\,,\\
z'_1(0)=0,\ z_1(x_0)=0\,,
\end{gather*}
 and 
\begin{gather*}
(|z_2'|^{p-2}z_2)'+Q_2(x)|z_2|^{p-2}z_2=0
\quad \text{on } [0,\pi_{p}-x_0]\,,\\
z'_2(0)=0,\ z_2(\pi_{p}-x_0)=0\,.
\end{gather*}
Let $\theta_1(x)$ and  $\theta_2(x)$ be the Pr\"ufer angles of
$z_1(x)$ and  $z_2(x)$ respectively. Then $\theta_1(x)$
and  $\theta_2(x)$ satisfy
\begin{gather*}
\theta_1'(x)=|S_{p}'(\theta_1(x))|^{p}+Q_1(x)|S_{p}(\theta_1(x))|^{p}\quad
\text{on } [0,x_0]\,,\\
\theta_2'(x)=|S_{p}'(\theta_2(x))|^{p}+Q_2(x)|S_{p}(\theta_2(x))|^{p}\quad
\text{on } [0,\pi_{p}-x_0]\,,\\
\theta_1(0)=\theta_2(0)=\frac{\pi_{p}}{2}\,,\\
\theta_1(x_0)=\theta_2(\pi_{p}-x_0)=\pi_{p}\,.
\end{gather*}
By comparison theorem, we find  $\pi_{p}-x_0\geq x_0$ and
hence $x_0\leq \pi_{p}/2$.
\end{proof}

\section{Two generalized triangular equations}

In this section, we will study the order of the roots of two
generalized triangular equations which are obtained
 from the proofs of Theorems \ref{thm1.1} and \ref{thm1.2} in section 4.
Define
\[
f(t)=t^{1/p}RT_{p}(t^{1/p}\frac{\pi_{p}}{2}),\quad
g(t)=t^{1/p}RT_{p}(t^{1/p}\frac{\pi_{p}}{2}+\frac{\pi_{p}}{2}).
\]
 We have the following results.

\begin{lemma}\label{lem3.1}
Let $m>0$. Let $t_1$ be the first root of $f(t)=-f(t-m)$ and
$t_2$ be the second root of $g(t)=-g(t-m)$. Then $t_2> t_1$.
\end{lemma}

\begin{proof}
First, note that $t_1\in (1,\min\{1+m, 2^{p}\})$ and $t_2\in (1, 3^{p})$ for $m>0$.

(i) Assume $t\geq 0$. Then, by Lemma \ref{lem2.1}, we find
\begin{align*}
g'(t)
&=\frac{1}{p}t^{\frac{1-p}{p}}RT_{p}(t^{1/p}\frac{\pi_{p}}{2}
 +\frac{\pi_{p}}{2})-t^{1/p}\big(1+|T_{p}(t^{1/p}\frac{\pi_{p}}{2}
 +\frac{\pi_{p}}{2})|^{p}\big)\\
 &\quad\times RT_{p}^{2}(t^{1/p}\frac{\pi_{p}}{2}
 +\frac{\pi_{p}}{2})\cdot\frac{1}{p}t^{\frac{1-p}{p}}\frac{\pi_{p}}{2}\\
&=\frac{t^{\frac{1-p}{p}}}{2p|S_{p}(t^{1/p}\frac{\pi_{p}}{2}
 +\frac{\pi_{p}}{2})|^{2}}\Big\{2S_{p}(t^{1/p}\frac{\pi_{p}}{2}
 +\frac{\pi_{p}}{2})S_{p}'(t^{1/p}\frac{\pi_{p}}{2}
 +\frac{\pi_{p}}{2})\\
&\quad -t^{1/p}\pi_{p}|S_{p}'(t^{1/p}
 \frac{\pi_{p}}{2}+\frac{\pi_{p}}{2})|^{2-p}\Big\}\\
&\equiv \frac{t^{\frac{1-p}{p}}}{2p|S_{p}(t^{1/p}\frac{\pi_{p}}{2}+\frac{\pi_{p}}{2})|^{2}}\tilde{g}(t).
\end{align*}
If $S_{p}'(t^{1/p}\frac{\pi_{p}}{2}+\frac{\pi_{p}}{2})
>0$, in this case $t^{1/p}\in (2+4n,4+4n)$ for $n\geq 0$,
then
\begin{align*}
\tilde{g}(t)
&= S_{p}'(t^{1/p}\frac{\pi_{p}}{2}+\frac{\pi_{p}}{2}) [2S_{p}(t^{1/p}\frac{\pi_{p}}{2}+\frac{\pi_{p}}{2})-t^{1/p}\pi_{p}|S_{p}'(t^{1/p}\frac{\pi_{p}}{2}+\frac{\pi_{p}}{2})|^{1-p}]\ \\
&\leq S_{p}'(t^{1/p}\frac{\pi_{p}}{2}) [2S_{p}(t^{1/p}\frac{\pi_{p}}{2})-t^{1/p}\pi_{p}]\\
&\equiv  S_{p}'(t^{1/p}\frac{\pi_{p}}{2}) h(t) .
\end{align*}
Since $h((2+4n)^{p})<0$ for $n\geq 0$, and
$h'(t)=\frac{t^{\frac{1-p}{p}}\pi_{p}}{p}(S_{p}'(t^{1/p}\frac{\pi_{p}}{2}+\frac{\pi_{p}}{2})-1)<0$
for  $t^{1/p}\in (2+4n,4+4n)$ and $n\geq 1$, we have
$h(t)<0$ for  $t^{1/p}\in (2+4n,4+4n)$ and $n\geq 0$.
Hence $g'(t)<0$ for $t^{1/p}\in (2+4n,3+4n)\cup
(3+4n,4+4n)$ and $n\geq 0$.

Similarly, if
$S_{p}'(t^{1/p}\frac{\pi_{p}}{2}+\frac{\pi_{p}}{2}) <0$,
in this case $t^{1/p}\in  (4n,4n+2)$ for $n\geq 0$, then
\begin{align*}
\tilde{g}(t)
&= S_{p}'(t^{1/p}\frac{\pi_{p}}{2}+\frac{\pi_{p}}{2}) [2S_{p}(t^{1/p}\frac{\pi_{p}}{2}+\frac{\pi_{p}}{2})+t^{1/p}\pi_{p}|S_{p}'(t^{1/p}\frac{\pi_{p}}{2}+\frac{\pi_{p}}{2})|^{1-p}] \\
&\leq S_{p}'(t^{1/p}\frac{\pi_{p}}{2}+\frac{\pi_{p}}{2}) [2S_{p}(t^{1/p}\frac{\pi_{p}}{2}+\frac{\pi_{p}}{2})+t^{1/p}\pi_{p}] \\
&\equiv S_{p}'(t^{1/p}\frac{\pi_{p}}{2}+\frac{\pi_{p}}{2})
\tilde{h}(t)\,.
\end{align*}
Since $\tilde{h}((4n)^{p})>0$ for $n\geq 0$ and
$\tilde{h}'(t)=\frac{t^{\frac{1-p}{p}}\pi_{p}}{p}(S_{p}'
(t^{1/p}\frac{\pi_{p}}{2}+\frac{\pi_{p}}{2})+1)>0$
 for $t^{1/p}\in (4n,4n+2)$ and $n\geq 0$,
 we have $\tilde{h}(t)>0$ for $t^{1/p}\in (4n,4n+2)$ and $n\geq 0$
and hence $g'(t)<0$ for $t^{1/p}\in  (4n,4n+1)\cup(4n+1,4n+2)$ and
$n\geq 0$.
\smallskip


(ii) Assume $t< 0$.
  Let $\hat{t}=-t$ and $\tilde{t}=\hat{t}^{1/p}\frac{\pi_{p}}{2}
+(-1)^{-1/p}\frac{\pi_{p}}{2}$. Since
$$
g(t)=t^{1/p}RT_{p}
(t^{1/p}\frac{\pi_{p}}{2}+\frac{\pi_{p}}{2})
=(-1)^{1/p}\hat{t}^{1/p}\frac{S_{p}'((-1)^{1/p}\tilde{t})}{S_{p}((-1)^{1/p}\tilde{t})}
=\hat{t}^{1/p}\frac{Sh_{p}'(\tilde{t})}{Sh_{p}(\tilde{t})}\,,
$$ 
we have
\begin{align*}
g'(t)
&= -\frac{1}{p}\hat{t}^{\frac{1-p}{p}}\frac{Sh_{p}'(\tilde{t})}{Sh_{p}(\tilde{t})}
+\hat{t}^{1/p}(-\frac{1}{p}\hat{t}^{\frac{1-p}{p}})
\frac{\pi_{p}}{2}\Big(\frac{Sh_{p}''(\tilde{t})}{Sh_{p}(\tilde{t})}
-\frac{Sh_{p}'^{2}(\tilde{t})}{Sh_{p}^{2}(\tilde{t})}\Big)\\
&= \frac{-\frac{1}{p}\hat{t}^{\frac{1-p}{p}}}{Sh_{p}^{2}(\tilde{t})}
\Big[Sh_{p}'(\tilde{t})Sh_{p}(\tilde{t})
+\frac{\pi_{p}}{2}\hat{t}^{1/p}\Big(\frac{|Sh_{p}(\tilde{t})|^{p}}{Sh_{p}'^{p-2}
(\tilde{t})}-Sh_{p}'^{2}(\tilde{t})\Big)
\Big]\\
&= \frac{-\frac{1}{p}\hat{t}^{\frac{1-p}{p}}}{Sh_{p}^{2}(\tilde{t})}
[ Sh_{p}'(\tilde{t})Sh_{p}(\tilde{t})-\frac{\pi_{p}}{2}
 \hat{t}^{1/p}Sh_{p}'^{2-p}(\tilde{t})]\\
&\equiv \frac{-\frac{1}{p}\hat{t}^{\frac{1-p}{p}}}{Sh_{p}^{2}(\tilde{t})}
\hat{g}(t)\,.
\end{align*}
Using similar argument as step (i), we can show $\hat{g}(t)>0$ and
hence $g'(t)<0$ for all $t<0$.
\smallskip

(iii) Let $t=t(m)$.
If $g(t)=-g(t-m)$, then
$g'(t)\frac{dt}{dm}=-g'(t-m)(\frac{dt}{dm}-1)$ and hence
$$0<\frac{dt}{dm}=\frac{g'(t-m)}{g'(t)+g'(t-m)}<1\,.$$ This
implies $t_2(m)$ is strictly increasing for $m>0$. On the other
hand, when $m=2^{p}$, we have $t_2=2^{p}$ and
$$t_2-t_1>0\quad \text{for}\ m\geq 2^{p}\,.$$ Theqrefore, we
only need to consider $0<m<2^{p}$. In this case, $t_1\in
(1,\min\{1+m,2^{p}\})$ and $t_2\in (\max\{1,m\},
\min\{2^{p},1+m\} )$.
\smallskip

(iv) Assume $t_1\geq t_2$ for some $0<m<2^{p}$. By similar arguments as steps 
(i) and (ii), it can be shown that $f(t)$ is decreasing on $(-\infty, 2^{p})$ 
and $((2n)^{p},(2n+2)^{p})$ for $n\geq 1$, and $f(1)=0$. Then
$$
-f(t_2-m)\leq -f(t_1-m)=f(t_1)\leq f(t_2)\leq f(t_2-m)\,.
$$
This implies $f(t_2-m)=0$ and then $t_2-m=1$. But $t_2< 1+m$. 
Hence $t_1< t_2$ for $m>0$.
\end{proof}


\begin{lemma}\label{lem3.2}
 Let $m>1$. Let $s_1$ be the first root of $f(s)=-f(sm)$ and
$s_2$ be the second root of $g(s)=-g(sm)$. Then $s_2> s_1$.
\end{lemma}

\begin{proof}
  Note that   $s_1, s_2\in (\frac{1}{m}, \min\{1, \frac{2^{p}}{m}\} )$. 
If $s_1\geq s_2$ for some $m>1$,
then $\frac{1}{m}\leq s_2<s_2m<2^{p}$ and 
$$
f(s_2)\geq f(s_1)=-f(s_1m)\geq -f(s_2m)>-f(s_2)\,.
$$ 
This implies $s_2=1$. Hence $s_1\leq s_2$ for $m>1$.
\end{proof}


\section{Proof of Main Theorems}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
For $M>0$, denote 
$$
A_{M}=\big\{ 0\leq q(x)\leq M : q
\text{ is single-well with transition point } \frac{\pi_{p}}{2}\big\}.
$$
Then $A_{M}$ is closed and  $E(q)\equiv (\nu_1-\mu_1)(q)$  is bounded on $A_{M}$.
Hence  there exists an optimal function $q_0$ giving the
minimal eigenvalue gap $\nu_1-\mu_1$.

Recall the definitions of $x_{-}$ and $x_{+}$ in \eqref{eq2.1}. We
shall define $q(x,t)=(1-t)q_0(x)+tq_1(x)$ for
$t\in [0,\pi_{p}]$ for some appropriated  function $q_1$.

First, assume  $x_{-}\leq \pi_{p}/2 \leq x_{+}$. Let
$$
q_1(x)=\begin{cases}
q_0(x_{-})&\text{on } (0,\frac{\pi_{p}}{2}),\\
q_0(x_{+})&\text{on } (\frac{\pi_{p}}{2},\pi_{p}).
\end{cases}
$$
 By the optimality of $q_0$ and Lemma \ref{lem2.3}, we have
\begin{align*}
0&\leq\ frac{d}{dt}(\nu_1(t)-\mu_1(t))|_{t=0}\\
&= \int_0^{\pi_{p}}(q_1(x)-q_0(x))(|\psi_1(x,0)|^{p}-|\phi_1(x,0)|^{p})dt\,,
\end{align*}
which is nonpositive. Hence, $q_0(x)=q_1(x)$ a.e. on
$[0,\pi_{p}]$.

If $x_{-}>\pi_{p}/2$, we let 
$$
q_1(x)=\{\begin{cases}
0& \text{on } (0,x_{-}),\\
M& \text{on } (x_{-},\pi_{p}).
\end{cases}
$$
By the normality
of $\phi_1$ and $\psi_1$, we have 
$$
\int_0^{x_{-}}(|\psi_1(x,0)|^{p}-|\phi_1(x,0)|^{p})dx > 0>
\int_{x_{-}}^{\pi_{p}}(|\psi_1(x,0)|^{p}-|\phi_1(x,0)|^{p})dx .
$$ 
Hence, by the optimality of $q_0$, we have
\begin{align*}
0&\leq \frac{d}{dt}(\nu_1(t)-\mu_1(t))|_{t=0}\\
&= \int_0^{\pi_{p}}(q_1(x)-q_0(x))(|\psi_1(x,0)|^{p}-|\phi_1(x,0)|^{p})dx\\
&= \int_0^{x_{-}}(-q_0(x))(|\psi_1(x,0)|^{p}-|\phi_1(x,0)|^{p})dx\\
&\quad + \int_{x_{-}}^{\pi_{p}}(M-q_0(x))(|\psi_1(x,0)|^{p}-|\phi_1(x,0)|^{p})dx\\
&\leq -q_0(\frac{\pi_{p}}{2})\int_0^{x_{-}}(|\psi_1(x,0)|^{p}
 -|\phi_1(x,0)|^{p})dt\\
&\quad +(M-q_0(x_{+}))\int_{x_{-}}^{\pi_{p}}(|\psi_1(x,0)|^{p}
 -|\phi_1(x,0)|^{p})dt\,,
\end{align*}
which is non-positive. This implies that $q_0=0$ on $(0,x_{-})$
and $=M$ on $(x_{-},\pi_{p})$.
But this makes a contradiction to Lemma \ref{lem2.4}. 
Hence this case is refuted. The case $x_{+}\leq \pi_{p}/2$ is similar.


After simplification, the optimal function $q_0$ is a $1$-step
function. Without loss of generality, let 
$$ 
q_0(x)=\begin{cases}
0&\text{on } (0,\frac{\pi_{p}}{2}),\\
m&\text{on } (\frac{\pi_{p}}{2},\pi_{p}).
\end{cases}
$$ 
By equating the corresponding ratio by $y'/y$ at $\pi_{p}/2$,
 $\nu_1$ is the second root of the functional equation
$\lambda^{1/p}RT_{p}(\frac{\pi_{p}}{2}\lambda^{1/p}+\frac{\pi_{p}}{2})
=-(\lambda-m)^{1/p}RT_{p}(\frac{\pi_{p}}{2}(\lambda-m)^{1/p}+\frac{\pi_{p}}{2})$,
and, similarly, $\mu_1$ is the first root of
$\lambda^{1/p}RT_{p}(\frac{\pi_{p}}{2}\lambda^{1/p})=-(\lambda
-m)^{1/p}RT_{p}(\frac{\pi_{p}}{2}(\lambda-m)^{1/p})$.
 Using Lemma \ref{lem3.1}, we obtain $\nu_1-\mu_1> 0$.

Finally, if the transition point $a$ is not $\pi_{p}/2$, we let
$$
q(x,t)=\begin{cases}
t&\text{on } [0,a],\\
0&\text{on } [a,\pi_{p}].
\end{cases}
$$
Since 
$\phi_1(x,0)=(p/\pi_{p})^{1/p}S_{p} (x)$,
$\psi_1(x,0)=(p/\pi_{p})^{1/p}S_{p}(x+\pi_{p}/2)$, and
$$
\int_0^{\pi_{p}/2}(|\psi_1(x,0)|^{p}-|\phi_1(x,0)|^{p})dx=0,
$$
we have
$$
\frac{d}{dt}(\nu_1(t)-\mu_1(t))|_{t=0}=\int_0^{a}(|\psi_1(x,0)|^{p}
-|\phi_1(x,0)|^{p})dx < 0\,,
$$
 when $0<a-\frac{\pi_{p}}{2}<<1$. Hence for small $t>0$,
$\nu_1(t)-\mu_1(t)<0$ when $0<a-\pi_{p}/2<<1$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
For $M>1$, denote 
$$
A_{M}=\big\{ \frac{1}{M}\leq \rho(x)\leq M : \rho\
\text{is single-barrier with transition point}\
\frac{\pi_{p}}{2}\big\}.
$$ 
Then there exists an optimal function $\rho_0$
giving the minimal eigenvalue ratio $\nu_1/\mu_1$.

Similar to the proof of Theorem \ref{thm1.1}  and by Lemma
\ref{lem2.4}, the cases $x_{+}<\pi_{p}/2$ and
$x_{-}>\pi_{p}/2$ are refuted by using suitable $\rho_0$'s.
Hence we have 
$x_{-}\leq \pi_{p}/2\leq x_{+}$ and 
$$
\rho_0(x)=\begin{cases}
\rho_0(x_{-})&\text{on }(0,\frac{\pi_{p}}{2}),\\
\rho_0(x_{+})&\text{on }(\frac{\pi_{p}}{2},\pi_{p}). 
\end{cases}
$$
That is the optimal function $\rho_0$ is a $1$-step function.
Without loss of generality, let 
$$
\rho_0(x)=\begin{cases}
1&\text{on } (0,\frac{\pi_{p}}{2}),\\
m&\text{on } (\frac{\pi_{p}}{2},\pi_{p}),
\end{cases}
$$ 
for some $m>1$. Then $\nu_1$ is the second root of
$$
RT_{p}(\frac{\pi_{p}}{2}\lambda^{1/p}+\frac{\pi_{p}}{2})
=-m^{1/p}RT_{p}(\frac{\pi_{p}}{2}(m\lambda)^{1/p}+\frac{\pi_{p}}{2}),
$$
and $\mu_1$ is the first root of
 $$
RT_{p}(\frac{\pi_{p}}{2}\lambda^{1/p})
=-m^{1/p}RT_{p}(\frac{\pi_{p}}{2}(m\lambda)^{1/p}).
$$
 Hence, by Lemma \ref{lem3.2},
$\nu_1/\mu_1>1$.

Finally, we let 
$$
\rho(x,t)=\begin{cases}
t&\text{on } [0,a],\\
1&\text{on } [a,\pi_{p}].
\end{cases}
$$
Then it can be shown that, if $0< \pi_{p}/2 -a<< 1$, the function
 $\rho(x,t)$ gives $\nu_1/\mu_1<1$ for small $t>1$.
\end{proof}

\subsection*{Acknowledgments}
The author is partially supported by Ministry of Science and Technology,
Taiwan, Republic of China, under contract nos. NSC 102-2115-M-152-002.


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\end{document}