\section{Introduction}\label{section 1}
Let us first recall the famous conjecture by Lord Rayleigh from his book `The theory of sound' \cite{Rayleigh} published in 1877. He conjectured that   
{\it `among all  planar domains $\Om$ of fixed area, the disk is the  domain that minimises the first Dirichlet eigenvalue $\la_1(\Om)$ of the Laplacian.'} This conjecture was open for a very long time. The first proof of this conjecture was published in 1923 by Faber \cite{Faber}. In 1925, Krahn \cite{Krahn}  gave an independent proof for this conjecture, and later he extended the same to the higher dimension.  Now this result is collectively known as the Faber-Krahn inequality, and it states that: 
\begin{equation}\label{F-K}
\la_1(\Om^*)\leq \la_1(\Om)
\end{equation} 
{\it with the equality if and only if $\Om$  is a ball (up to a measure zero set), where $\Om^*$ is a ball of the same measure as $\Om$.}  For more on Faber-Krahn inequality and related results, we refer to \cite{Payne} and \cite{Polya-Szego}.

Among the class of domains having a fixed volume, $\la_1(\Om)$ is not bounded above. For example, consider a class of arbitrary long rectangular domains with the same volume (see Remark 3.13 of \cite{Anoop}). However, if we fix both the surface area and the volume one can give upper bounds for $\la_1(\Om )$, see \cite{Makai1} and \cite{Polya}. In \cite{P-W} Payne-Weinberger considered  this problem for bounded planar membrane with fixed  outer boundary having the interior boundaries along which it is free. They showed that `{\it among the class of membranes with given area $A$ and given length $L_0$ of the outer boundary,  the annulus $\Om^\#$ has the highest fundamental frequency,}' where $\Om^\#$ is a concentric annulus with the same area as $\Om$ and the same outer boundary length as $L_0$. In other words, this result can be stated in the form of an inequality as below:
\begin{equation}\label{Rev1}
\nu_1(\Om) \le  \nu_1(\Om^\#),
\end{equation} where $\nu_1(\Om),\, \nu_1(\Om^\#)$ are first eigenvalues of the following mixed eigenvalue problems:\\
\begin{minipage}[t]{0.49\textwidth}
	\begin{equation*}
	\left.
	\hs{-0.8cm}
	\begin{aligned}
	-\De u&= \nu u \quad \text{in } \, \Om,\\
	u&=0 \text{ on }  \Ga_0,\\
	\frac{\pa{u}}{\pa \eta}&=0  \text{ on }   \pa\Om\setminus\Ga_0;
	\end{aligned}
	\right\}
	\end{equation*}\hspace*{3cm}
\end{minipage}
\begin{minipage}[t]{0.49\textwidth}
	\begin{equation*}
	\left.
	\hs{-0.8cm}
	\begin{aligned}
	-\De u&= \nu u \quad \text{in } \, \Om^\#,\\
	u&=0 \text{ on }  \Ga_0^\#,\\
	\frac{\pa{u}}{\pa \eta}&=0  \text{ on }   \pa\Om^\#\setminus \Ga_0^\#,
	\end{aligned}
	\right\}
	\end{equation*}
\end{minipage}\\
where $\Ga_0$ and  $\Ga_0^\#$ are the outer boundaries of $\Om$ and $\Om^\#$ respectively.

We call the inequality in \eqref{Rev1}, as the reverse Faber-Krahn (R-F-K) inequality. A  similar inequality for the second Neumann eigenvalue (the first non-zero eigenvalue), namely $\mu_2(\Om)\le \mu_2(\Om^*)$ is obtained in  \cite{Szego} by Szeg\"o  for planar domains, and  in \cite{Weinberger-Neumann} for  higher dimensions by Weinberger. The proof of Szeg\"o mainly rely on the conformal mapping, and the proof of Weinberger based on construction of a test function using a radial function together with a suitable translation of the origin. In \cite{P-W}, for  proving the inequality \eqref{Rev1}, authors used the interior parallels and an isoperimetric inequality, deduced from an inequality for the  interior parallels due to  B. Sz. Nagy \cite{Nagy}. Next, we briefly describe the interior parallels and Nagy's inequality.

\noi{\bf The parallel sets:}  Let $\Om$ be a bounded domain in $\R^2$ and $\Ga_0$ be its outer boundary. Let $L(\de)$   be the measure of the inner parallel to $\Ga_0$-{\it  the set of all points that are inside $\Ga_0$ and at a distance $\de$ from $\Ga_0$.} Nagy showed that the function $L(\de)$ is defined for almost every $\de$ and  it satisfies 
\begin{equation}\label{Ineq:Nagy}
L(\de) \le |\Ga_0|-2\pi \de.
\end{equation}
Similarly, if $L(\de)$ is the measure of the outer parallel curve to $\Ga_1$, an interior boundary; he showed that $L(\de)$ satisfies 
\begin{equation}\label{Ineq2:Nagy}
L(\de) \le |\Ga_1|+2\pi \de.
\end{equation}

\noi Recall the Steiner formula $L(\de)=|\Ga_1|+2\pi \de$ for the convex planar domains, (see Chapter 4 of \cite{Schneider}). Indeed, Nagy's second inequality is an extension of Steiner formula for the non-convex planar domains. 

The {\it interior parallels} are the part of the parallels that lies in $\Om.$ If $l(\de)$ is the measure of the interior parallel at a distance $\de$, then $l(\de)\le L(\de).$ The interior parallels and Nagy's inequalities were used in \cite{Makai2}, \cite{Makai1} and \cite{Polya} for finding the upper bounds for $\la_1(\Om)$ among the class of domains having a fixed volume and surface area. In \cite{P-W}, using Nagy's inequality \eqref{Ineq:Nagy},  authors derived the following isoperimetric inequality:
\begin{equation}\label{P-W ineq}
l(\de)^2\leq |\Ga_0|^2-4\pi v(\de),
\end{equation} 
\noi where $v(\de)=\int_0^\de l(s)d\ds.$ Further, using  $v(\de)$  and the first eigenfunction of the concentric domain $\Om^\#$, they have constructed a test function on $\Om$ whose level sets coincide with interior parallels and the Rayleigh quotient is smaller than $\nu_1(\Om^\#).$  In \cite{Hersch}, Hersch gave another proof for \eqref{Rev1} using a class of test functions known as `web functions' (the test function that depends only on the distance from $\Ga_0$),  for more on web functions see \cite{BCT} and \cite{Gazzola} and the references therein. 

Hersch also considered the mixed eigenvalue problem with the Dirichlet condition  on an interior boundary and with the Neumann condition on the rest of the boundaries. By an ingenious parametrization $t(\de)=\int_0^\de \frac{\ds}{l(s)}$, he related this problem with the fundamental frequency of a vibrating string of infinite length (with varying mass distribution) fixed at one end. Then he used Nagy's inequality \eqref{Ineq2:Nagy} to show that `{\it among the class of membranes with given area $A$ and given length $L_1$ of an inner boundary $\Ga_1$, the annulus $\Om_\#$ has the maximal fundamental frequency,'} where $\Om_\#$ is a concentric annulus with the same area as $\Om$ and the same inner boundary length as $L_1$. As before, this can be restated as a R-F-K inequality:
\begin{equation}\label{Rev2}
\tau_1 (\Om) \le  \tau_1(\Om_\#),
\end{equation} where $\tau_1(\Om),\, \tau_1(\Om_\#)$ are first eigenvalues of the following problems:\\
\begin{minipage}[t]{0.49\textwidth}
	\begin{equation*}
	\left.
	\hs{-0.8cm}
	\begin{aligned}
	-\De u&= \tau u \quad \text{in } \, \Om,\\
	u&=0 \text{ on }  \Ga_1,\\
	\frac{\pa{u}}{\pa \eta}&=0  \text{ on }   \pa\Om\setminus\Ga_1;
	\end{aligned}
	\right\}
	\end{equation*}\hspace*{3cm}
\end{minipage}
\begin{minipage}[t]{0.49\textwidth}
	\begin{equation*}
	\left.
	\hs{-0.8cm}
	\begin{aligned}
	-\De u&= \tau u \quad \text{in } \, \Om_\#,\\
	u&=0 \text{ on }  \Ga_1^\#,\\
	\frac{\pa{u}}{\pa \eta}&=0  \text{ on }   \pa\Om_\#\setminus \Ga_1^\#.
	\end{aligned}
	\right\}
	\end{equation*}
\end{minipage}

\noi The mixed eigenvalue problem with the Dirichlet condition on both $\Ga_1,\Ga_0$, and with Neumann condition on the rest of the boundaries also studied in \cite{Hersch}. In this case, by an effective use of  the ``effectless cut'' due to Weinberger \cite{Weinberger} together with \eqref{Rev1} and \eqref{Rev2}, he showed that `{\it among the class of membranes with given area $A$, given length $L_0$ of the inner boundary $\Ga_0$ and given length $L_1$ of the outer boundary $\Ga_1$ satisfying $L_1^2- L_0^2=4\pi A$, the annulus has the highest fundamental frequency $\la_1$}.' In particular, an annulus $\Om=B_1\setminus \overline{B_0} \subset \R^2$  satisfies this relation. Thus by  Hersch's result:
\begin{equation}\label{Rev3}
\la_1 (B_1\setminus \overline{B_0}) \le  \la_1(B_1^*\setminus \overline{B_0^*}).
\end{equation}

In \cite{Ramm-Shivakumar}, Ramm and Shivakumar  conjectured the inequality  \eqref{Rev3} for $N\ge 3$,  with a numerical justification. In fact, they have conjectured a stronger result - \textit {`$\la_1(B_1\setminus \overline{B_0})$ strictly decreases when the inner ball moves towards the outer boundary.'} An  analytic  proof  for this  conjecture using an argument of M. Ashbaugh was   published later in an arxiv paper (arxiv:math-ph/9911040) by the same authors.  At the same time,    Harrell et al. \cite{HarrelKroger} and Kesavan  \cite{Kesavan} independently proved the strict monotonicity of $\la_1(B_1\setminus\overline{B_0})$. All the  proofs are based mainly  on the Hadamard perturbation formula. In  \cite{Anisa2}, the authors  studied   the monotonicity   of   the first Dirichlet eigenvalue of $p$-Laplacian, defined by $\Dep u =\mbox{div}(|\nabla u|^{p-2}\nabla u)$ for $p\in(1,\infty)$. They showed that \textit {`$\la_1(B_1\setminus \overline{B}_0)$  decreases when the inner ball $B_0$ moves towards the boundary of the outer ball $B_1$.'} Indeed, this result imply \eqref{Rev3} for the $p$-Laplacian. The strict monotonicity  is obtained in \cite{Anoop} and this result asserts that the equality happens in \eqref{Rev3} only if the balls $B_1$ and $B_0$ are concentric.


In this article, we study the R-F-K inequalities  \eqref{Rev1} and \eqref{Rev2} for the $p$-Laplacian with $p\in(1,\infty)$ and $N\ge 2$. We consider $\mathcal{C}^1$-smooth multiply connected bounded domains in $\R^N$.  For such a domain $\Om$,  we denote the outer boundary by $\Ga_0$ and the boundaries of the interior holes by $\Ga_1, \ga_1,\ldots \ga_n$. We use the symbol $|\,\cdot\,|$ for both $N$ dimensional volume measure and the $(N-1)$-dimensional surface measure. The Lebesgue measure of the unit ball in $\RN$ is denoted by $\om_N.$
Further, we set 


\begin{enumerate}[align=left,]
	\item[$\Om^*=$]  the  ball centred at the origin  with $|\Om^*|=|\Om|,$
	\item[$\Om^\#=$] the concentric annulus centred at the origin  with the same volume as $\Om$ and the outer surface measure as $\Ga_0,$
	\item[$\Om_\#=$] the concentric annulus centred at the origin  with the same volume as $\Om$
	and the inner surface measure as$\Ga_1.$
\end{enumerate}


In general, for a fixed domain $\Om$ the sets $\Om^\#$ and $\Om_\#$ are different. However, they coincide if $\Om$ satisfies the relation $|\Ga_0|^{\pr N}-|\Ga_1|^{\pr N}=C(N)|\Om|,$ where  $N^{\prime}$ is the H\"{o}lder conjugate of $N$ and $C(N)= N^{N^{\prime}} \om_N^{N^{\prime}-1}.$ We mainly consider two types of  mixed eigenvalue problems:  
 \begin{enumerate}[(i)]
\item the Dirichlet condition  on the outer boundary $\Ga_0$,
\item the Dirichlet condition  on an inner boundary $\Ga_1$,
\end{enumerate}
and the Neumann condition  on the rest of the boundaries. More precisely, for $p\in(1,\infty)$ we consider the following eigenvalue problems:

\begin{minipage}[t]{0.47\textwidth}
	\begin{equation}\label{ND}\tag{N-D}
	\left.
	\hs{-0.8cm}
		\begin{aligned}
	 -\De_p u&= \nu |u|^{p-2}u \quad \text{in } \, \Om,\\
	 u&=0 \text{ on }  \Ga_0,\\
	 \frac{\pa{u}}{\pa \eta}&=0  \text{ on }   \Ga_1\cup\left(\cup_{i=1}^n\ga_i\right);
	\end{aligned}
	\right\}
	\end{equation}\hspace*{3cm}
	\end{minipage}
	\begin{minipage}[t]{0.47\textwidth}
	\begin{equation}\label{DN}\tag{D-N}
	\left.
	\hs{-0.8cm}
		\begin{aligned}
	 -\De_p u&= \tau |u|^{p-2}u \quad \text{in } \, \Om,\\
	 u&=0 \text{ on }  \Ga_1,\\
	 \frac{\pa{u}}{\pa \eta}&=0  \text{ on }   \Ga_0\cup\left(\cup_{i=1}^n\ga_i\right).
	\end{aligned}
	\right\}
	\end{equation}
	\end{minipage}
	
	 \noi For $i=0,1,$ let 	$W_{\Ga_i}\dq\left\{u\in \wp : u|_{\Ga_i}=0 \right\}.$
	We say a real number $\nu$ is an eigenvalue of \eqref{ND}, if there exists $u\in W_{\Ga_0}\setminus\{0\}$ such that
	\begin{equation*}
	  \int_\Om |\nabla u|^{p-2}\nabla u\cdot \nabla v \dx = \nu \int_\Om |u|^{p-2}uv\dx,\, \forall v\in W_{\Ga_0}.  
	\end{equation*}
\noi	Similarly a real number $\tau$ is an eigenvalue of \eqref{DN}, if there exists $u\in W_{\Ga_1}\setminus\{0\}$ such that 
	\[\int_\Om |\nabla u|^{p-2}\nabla u\cdot \nabla v \dx = \tau \int_\Om |u|^{p-2}uv\dx,\, \forall v\in W_{\Ga_0}. \]
	The corresponding nonzero solutions are called the eigenfunctions associated to these eigenvalues. 
	Now consider
	\[J(u)\dq{\int_{\Om}\md{\nabla u}^p}\dx, \mbox{ and }\S\dq\left\{v\in \wp:\int_\Om |u|^p\dx=1\right\}.\]
	 For $i=0,1,$ one can easily see that the critical values of $J$  on $\S\cap W_{\Ga_i}$  are precisely the eigenvalues of \eqref{ND} and \eqref{DN}  respectively.  Thus  the classical Ljusternik-Schnirrelman theory, ensures
	 the existence of infinitely many eigenvalues  for both \eqref{ND} and \eqref{DN}, see Proposition \ref{propo:appendix 1}. 
	  In particular, the first eigenvalues  have the following variational characterisation:
	
	
	
	
	\[\nu_1(\Om)=\inf\limits_{u\in \S\cap W_{\Ga_0}} J(u),\ \tau_1(\Om)=\inf\limits_{u\in\S\cap W_{\Ga_1}} J(u).\]

	Both $\nu_1(\Om)$ and $\tau_1(\Om)$  are simple and the first eigenfunctions are of constant sign.  Moreover, the eigenfunctions corresponding to $\nu_1(\Om^\#)$  and $\tau_1(\Om_\#)$ are radial (see Appendix \ref{Appendix A}).
	
  The R-F-K inequalities, except \eqref{Rev3} for the annular regions, unfortunately did not get any attention for the domains in the higher dimensions and also for  the operators different from the Laplacian. Unlike in the case of Faber-Krahn inequality,  the uniqueness of the domains for which the equality holds in \eqref{Rev1} (similarly in \eqref{Rev2}) are not well understood, even for the planar domains. For $N=2,$ we extend the results of Payne-Weinberger and Hersch for the $p$-Laplacian. For $N\ge 3$, under an additional assumption on the boundary of $\Om,$ we  prove the R-F-K inequalities. To state this additional assumption, we make the following definition:

 \noi{\bf Definition:}
 Let $A,B$ be measurable sets in $\RN$. We say that $A$ is a $\mu$-translate of $B$, if there exists $x\in \R^N$ such that $|(A+x)\bigtriangleup B|=0,$ where $\bigtriangleup$ is the  symmetric difference of sets.

Now we state the R-F-K inequality for $\nu_1$: 
\begin{theorem}\label{thm 1}
Let $p\in(1,\infty)$ and $\Om\subset \R^N$ with $N\ge 3$ be a $\C^1$-smooth multiply connected bounded domain. If the outer boundary $\Ga_0$ of $\Om$ be  a sphere, then 
\begin{enumerate}[(i)]
 \item $ \nu_1(\Om) \le  \nu_1(\Om^\#),$
 \item if equality happens then $\Om$ is a $\mu$-translate of $\Om^\#$.
\end{enumerate}
\end{theorem}
This theorem generalises the result of Payne-Weinberger \cite{P-W}, and proves that `{\it among the class of domains with given measure, free along the interior boundaries and fixed along a sphere of given radius as the outer boundary, the concentric annulus $\Om^\#$ has the highest fundamental frequency}'. For example, see below  Figure \ref{figure}: 
\begin{figure}[h]\label{figure}
    \centering
    \includegraphics[width=0.75\textwidth]{dm.eps}
    \caption{The domains $\Om_1, \Om_2$ and $\Om^\#$ have the same measure  and the same outer boundary (up to a translation). Then  $\nu_1(\Om^\#)$ is strictly greater than $\nu_1(\Om_i).$}
    \end{figure}

       \noi 
       In general, the domain can have any finite number of  interior boundaries. Comparing the values of $\nu_1(\Om_i)$ with the number of interior boundaries is also an interesting problem. In a subsequent article, using the shape derivative  technique,  we also prove that the  $\nu_1(B_1\setminus B_0)$ strictly decreases when the inner ball $B_0$ moves towards the boundary  of  the outer  ball $B_1.$  

 We give two proofs for the above theorem. The first one mainly use some ideas from  \cite{P-W} and an isoperimetric inequality  of the following form:
\begin{equation}
s(\de)^{\pr N}\leq |\Ga_0|^{\pr N} - C(N)v(\de).
\end{equation} Even though Nagy's inequality is not available for higher dimensions, our assumption on  $\Ga_1$  allows us to derive the above isoperimetric inequality. In the second proof, we borrow some ideas from \cite{Hersch}  and give a proof without using the isoperimetric inequality. We feel that the proofs of Payne-Weinberger and Hersch  need more appreciation and may be applicable for other related problems; this is the reason for presenting two proofs for Theorem \ref{thm 1}.

Next we state the  R-F-K inequality for $\tau_1$: 
\begin{theorem}\label{thm 2}
Let $p\in(1,\infty)$  and $\Om$ be as in Theorem \ref{thm 1}. If the inner boundary $\Ga_1$ of $\Om$ be a sphere, then
\begin{enumerate}[(i)]
 \item $ \tau_1(\Om) \le  \tau_1(\Om_\#),$
 \item if equality happens then $\Om$ is a $\mu$-translate of $\Om_\#$.
\end{enumerate}
\end{theorem}
For proving the above result, we introduce a parametrization $t$, analogous to the one in \cite{Hersch}  as
\[t(\de)=\int_0^\de \displaystyle\frac{\dr}{s(r)^{\pr{p}-1}}.\] This parametrization and the eigenfunction corresponding to $\tau_1(\Om_\#)$ helps us to construct a test function on $\Om$, whose Rayleigh quotient is smaller than $\tau_1(\Om_\#).$ 

For $N=2$, as  we have Nagy's inequality for a general multiply connected bounded domain, we extend the results of Payne-Weinberger and Hersch for the $p$-Laplacian with $p\in(1,\infty).$ 
\begin{theorem}\label{thm 3}
Let $p\in(1,\infty)$ and $\Om\subset \R^2$  be a $\C^1$-smooth multiply connected bounded domain.  Then
  \[(i)\ \nu_1(\Om) \le  \nu_1(\Om^\#)\quad \text{and}\quad  (ii)\ \tau_1(\Om) \le  \tau_1(\Om_\#).\]
\end{theorem}

\textbf{The second eigenvalues:}
In \cite{Anoop-Drabek}, Anoop et al. proved that the eigenfunctions corresponding to the second Dirichlet eigenvalue of $p$-Laplacian are non-radial. As an application of above theorems, we prove the similar results for the second eigenvalues of $p$-Laplacian on a ball or an annulus with various boundary conditions. 
Let $\mu_2(\Om)$ be the second (first non-zero) Neumann eigenvalue of $p$-Laplacian; let $\nu_2(\Om)$ and $\tau_2(\Om)$ be the second eigenvalues of \eqref{ND} and \eqref{DN} respectively. 
\noi Now, we have the following results.
\begin{theorem}\label{thm 4}
	For $N\geq 2$ and $1<p<\infty$, let $\Om\subset\RN$ be a ball or a concentric annulus in $\RN$. Then the nodal set of any eigenfunction associated with $\mu_2(\Om)$ cannot be a concentric sphere.
\end{theorem}


\begin{theorem}\label{thm 5}
		For $N\geq 2$ and $1<p<\infty$, let $\Om\subset\RN$ be a concentric annulus. Then the nodal set of any eigenfunction associated with $\nu_2(\Om)$ and $\tau_2(\Om)$ cannot be a concentric sphere. 
\end{theorem}


This article is organised as follows. In the first section, we  briefly illustrate  the interior parallels  and derive Payne-Weinberger's type isoperimetric inequality. The proofs for Theorem \ref{thm 1}, Theorem \ref{thm 2} and Theorem \ref{thm 3} are also given in this section. In the next section, we give the proofs of Theorem \ref{thm 4} and Theorem \ref{thm 5}. In Section \ref{section 4}, we  remark on a similar result for the elasticity problem and state a few open problems related to the mixed  eigenvalue problems. In the appendix, we discuss some properties of the first eigenfunctions and give the variational characterisation of the second eigenvalues.
{"config": "arxiv", "file": "1806.11491/intro.tex"}