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\begin{document}

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\title{Mixed Eigenvalues of Discrete {$\pmb p$\,}-Laplacian}

\author[Yu-Hui Zhang]{Yu-Hui Zhang}

\date[July 1, 2013]{\wuhao IMS-SWUFE International Conference on
 \\ Statistics and Probability(June 30-July 4, 2013) \\
$ $\\ This talk is based on the joint work with Mu-Fa Chen and
Ling-Di Wang.}

\institute[BNU]{\sihao Beijing Normal University}

\frame{\titlepage}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%            目录            %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{目录}
  %  \frame{\frametitle{目录}\tableofcontents}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%         背景及问题        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\large
%----------------------------------------------------------------------------------------1

\begin{frame}
 \frametitle{Outline}
\vskip0.5cm
\begin{enumerate}
\item Problem
\vskip0.3cm
\item Background
\vskip0.3cm
\item Our work\\\small{
\qquad $(1)$ Analysis \& Notations\\
\qquad $(2)$ Variational formulas\\
\qquad $(3)$ Basic estimates \\
\qquad $(4)$ Approximating procedure}
\vskip0.3cm
\item Examples
\end{enumerate}
\end{frame}

%----------------------------------------------------------------------------------------2
\frame{ \frametitle{1. Problem}\small
\begin{itemize}
\item<-1>{\color{blue} the discrete version of $p$-Laplacian eigenvalue problem:}
\begin{eqnarray*}
\text{Eigeneq.:}& \Omega_pg(k)=-\lambda\mu_k|g_k|^{p-2}g_k,\ k\in E:=\mathbb{Z}^+\cap[0, N],\end{eqnarray*}
$$\aligned \Omega_pg(k):=&\nu_{k-1}|g_{k-1}-g_k|^{p-2}(g_{k-1}-g_k)\\
&-\nu_k|g_k-g_{k+1}|^{p-2}(g_k-g_{k+1}).\endaligned$$
\item<-1>{\color{blue}ND boundary conditions~($N\leqslant\infty$)}
$$ 0\ne g_{-1}=g_0~\text{and}~ g_{N+1}=0~\text{if}~ N<\infty.$$
\end{itemize}
\vskip-1cm
\begin{block}{Problem}
\qquad How to estimate on the first eigenvalue $\lambda_p(1<p<\infty)$?
\end{block}
}
%----------------------------------------------------------------------------------------3
\frame{\frametitle{2. Background }%\small
$p=2$, DD, DN, ND and NN
\begin{itemize}
\item Chen M F. Speed of stability for birth-death process, Front. Math. China, 2010, 5(3): 379-516.
\item Chen M F, Wang L D, Zhang Y H. Mixed principal eigenvalues in dimension one, Front. Math. China, 2013, 8(2): 37-343.
 \end{itemize}
$p\ne2$, NN and DD
\begin{itemize}
\item Chen M F. Bilateral Hardy-type inequalities, Acta Math. Sinica, 2013, 29(1): 1-32.
\end{itemize}
}
%----------------------------------------------------------------------------------------4
\frame{\small
\begin{itemize}
\item the classical variational formula:
$$\lambda_p=\inf\big\{D_p(f):\mu\big(|f|^p\big)=1, f\in \mathscr{C}_K\big\},$$
where $\mathscr{C}_K$ is the set of functions having compact support and
$$D_p(f):=\sum_{k\in E}\nu_k|f_k-f_{k+1}|^p,\quad p\geqslant1,\quad f\in \mathscr{C}_K.$$
\item the weighted Hardy inequality:
$$\mu\big(|f|^p\big)\leqslant A D_p(f),\quad f\in\mathscr{C}_K$$
\item the optimal constant $A=\lambda_p^{-1}$. This explains the relationship between the $p$-Laplacian eigenvalue and the Hardy's inequality.
 \end{itemize}

}
%----------------------------------------------------------------------------------------5
\frame{\frametitle{3(1). Our work: \small{Analysis \& Notations}}\small
$$\aligned &\text{eigenequation}\quad+\quad\text{ND boundary conditions}\\
&{\green\Longrightarrow}\nu_j|g_j-g_{j+1}|^{p-2}(g_j-g_{j+1})=\lambda_p\sum_{k=0}^j\mu_k|g_k|^{p-2}g_k.\\&
{\green\big\Downarrow}{\red\text{if}~ g~\text{monotone}}\Longrightarrow{\blue I_i(g):=\frac{1}{\nu_i(g_i-g_{i+1})^{p-1}}\sum_{j=0}^i\mu_jg_j^{p-1}}=\lambda_p^{-1}\\&
 g_{n}-g_{N+1}=\lambda_p^{p'-1}\sum_{j=n}^{N}\bigg(\frac{1}{\nu_j}\sum_{k=0}^{j}\mu_k|g_k|^{p-2}g_k\bigg)^{p'-1} \\
&{\green\big\Downarrow} {\red \text{if} ~g_{N+1}=0}\Rightarrow
 {\blue I\!I_n(g):=\bigg\{\frac{1}{g_n}\sum_{j=n}^{N}\bigg(\frac{1}{\nu_j}\sum_{k=0}^{j}\mu_k|g_k|^{p-2}g_k\bigg)^{p'-1}\!\bigg\}^{p-1} }\!\!\!\!\!=\lambda_p^{-1}\endaligned$$
}

%--------------------------------------------------------------------6
\frame{%\small
\begin{itemize}
\item For the eigenfunction $g$ of $\lambda_p$, then 
$$\blue{ \lambda_p=I_i(g)^{-1}=I\!I_i(g)^{-1}}.$$
\end{itemize}
\begin{block}{About the eigenfunction $g$}
\begin{itemize}
\item[$(1)$]  $g$ is either positive or negative.
\item[$(2)$]  $g$ is strictly monotone.
\item [$(3)$] $g_{N+1}:=\lim_{i\to N+1}g_i=0$ for $N=\infty$.
\end{itemize}
\end{block}
}

%----------------------------------------------------------------------------7
\begin{frame}
Based the properties of eigenfunction, the set of test functions is defined as follows:
\begin{itemize}
\item for lower bounds $\mathscr{F}_{I\!I}=\{f:\ f>0 \}$.
\item for upper bounds
$$
\aligned
\widetilde{\mathscr{F}}_{I\!I}=\{f:f_i>0&~\text{up to some}~m\in[1,N+1)\\
& \text{and then vanishes}\}.
\endaligned
$$
\item{\blue$$\aligned\inf_{f\in\widetilde{\mathscr{F}}_{I\!I}}\sup_{i\in {\rm supp}(f)}I\!I_{i}(f)^{-1}=\lambda_p=\sup_{f\in\mathscr{F}_{I\!I}}\inf_{i\in E}I\!I_{i}(f)^{-1}.
\endaligned$$}
\end{itemize}
\end{frame}

%----------------------------------------------------------8
\begin{frame}
\frametitle{3(2). Variational formulas}\small
\begin{block}{Theorem 1 (Variational formulas)}
(1)Single summation form:
$$\aligned
\inf_{f\in\widetilde{\mathscr{F}}_{I}}\sup_{i\in E}I_i(f)^{-1}=\lambda_{p}=\sup_{f\in\mathscr{F}_{I}}\inf_{i\in E}I_i(f)^{-1}.
\endaligned$$
(2) Double summation form:
$$\aligned
\inf_{f\in\widetilde{\mathscr{F}}_{I\!I}}\sup_{i\in \rm supp(f)}I\!I_i(f)^{-1}=\lambda_{p}=\sup_{f\in\mathscr{F}_{I\!I}}\inf_{i\in E}I\!I_i(f)^{-1},
\endaligned$$
(3) Difference form:
$$\aligned
\inf_{w\in\widetilde{\mathscr{W}}}\sup_{x\in E}R_i(w)=\lambda_{p}=\sup_{w\in\mathscr{W}}\inf_{i\in E}R_i(w).
\endaligned$$
Moreover, the supremun on the right-hand side of the three above formulas can be attained.
\end{block}
\end{frame}
%----------------------------------------------------------9
\begin{frame}
\small
 $$\aligned
&\mathscr{F}_{I}=\{f: f>0 \text{ and } f  \text{ is strictly decreasing}\},\\
&\widetilde{\mathscr{F}}_{I}=\{f: f \ \text{is strictly decreasing on some}\, [n,m] , 0\leqslant n<m<N+1, \\
& \qquad \qquad \;\; f_{\cdot}=f_{\cdot\vee n}\mathbbold{1}_{.\leqslant m}\},\\
&\mathscr{W}=\bigg\{w: w_{i}\in(0,1)\text{ if } \sum_{j\in E}\hat{\nu}_j<\infty\text{ and } w_{i}\in(0,1] \text{ if } \sum_{j\in E}\hat{\nu}_j=\infty\bigg\},\\
&\widetilde{\mathscr{W}}=\Big\{w: \exists\,  m\in[1,N+1)\text{ such that }  w_{i}>0 \text{ up to } m-1,  w_{m}=0, \\&\hskip1.8cm w_{i}<1-{(\nu_{i-1}/\nu_i)^{p-1}}\big(w_{i-1}^{-1}-1\big)\text{ for } i=0,1,\cdots,m\Big\}.
\endaligned
$$

$$R_{i}(w)=\mu_i^{-1}\big[\nu_i(1-w_{i})^{p-1}-\nu_{i-1}\big(w_{i-1}^{-1}-1\big)^{p-1}\big]\quad(\text{difference form}).$$

\end{frame}


%----------------------------------------------------------10
\begin{frame}
\frametitle{3(3). Basic estimates}\small
Applications of variational formulas:
\begin{block}{Proposition}
Assume that $\lambda_p>0$. Then, for the eigenfunction $g$ of $\lambda_p$, $g_{N+1}:=\lim_{i\rightarrow N+1}g_{i}=0$ for $N=\infty$.
\end{block}
Define

 $\hat{\nu}_j=\big({\nu_j}\big)^{1-p'}$, $k(p)=pp'^{p-1}$, $\varphi_n=\hat{\nu}[n,N]^{p-1}$, $ \sigma_p=\sup_{n\in E}\mu[0,n]\varphi_n$
\begin{block}{Theorem 2 (Basic estimates)}
 $$\lambda_p>0\Leftrightarrow\sigma_p<\infty.$$
More precisely,
$$(k(p)\sigma_p)^{-1}\leqslant\lambda_p\leqslant\sigma_p^{-1}.$$
\end{block}
\end{frame}
%----------------------------------------------------------11
\begin{frame}
\small
\begin{itemize}
\item  Kufner A. et al. 2007. The Hardy Inequality, Theorem 7, P.58;  Mao Y H, Nash ineq., 2002.
\item  Kufner A, Persson L E. 2003, Weighted Hardy Inequality.
\end{itemize}
\begin{figure}[h]
\begin{center}{\includegraphics[width=6.5cm,height=4.8cm]{Hardy-type-0.eps}}\end{center}
The function $p\to k(p)^{1/p}$ is unimodal with maximum $2$ at $p=2$.
\end{figure}
\end{frame}
%----------------------------------------------------------12
\begin{frame}
\frametitle{3(4). Approximating procedure}\small
\begin{block}{Theorem 3 (Approximating procedure)} Assume that $\sigma_p<\infty$.
\begin{itemize}\setlength{\itemsep}{-0.8ex}
\item[$(1)$]Define $f_1=\varphi^{1/p}$, {\blue $f_n=f_{n-1}I\!I(f_{n-1})^{p'-1}$, $\delta_n=\sup_{i\in E}I\!I_i(f_n)$}. Then {\blue $\delta_n $ $\downarrow$} and
$$\lambda_p\geqslant \delta_{\infty}^{-1}\geqslant\cdots\geqslant\delta_1^{-1}\geqslant(k(p)\sigma_p)^{-1}.$$
\item[$(2)$] Define $\begin{displaystyle}\delta_n'=\sup_{\ell,m: \ell<m}\min_{i\leqslant m}I\!I_i(f_n^{(\ell,m)}).\end{displaystyle}$
Then {\blue$\delta_n'$ $\uparrow$} and
$$\sigma_p^{-1}\geqslant\delta_1'^{-1}\geqslant\cdots\geqslant\delta_{\infty}'^{-1}\geqslant\lambda_p.$$
\item[$(3)$] Define $\begin{displaystyle}\bar{\delta}_n=\sup_{\ell,m: \ell<m}\frac{\mu(|f_n^{(\ell,m)}|^p)}{D_p(f_n^{(\ell,m)})}.\end{displaystyle}$ Then {\blue$\bar{\delta}_n^{-1}\geqslant\lambda_p$, $\bar{\delta}_{n+1}\geqslant\delta_n'$.}
\end{itemize}
\end{block}
\end{frame}
%----------------------------------------------------------13
\begin{frame}
\small
\begin{block}{Corollary (Improved estimates)}
      \small{
$$\sigma_p^{-1}\geqslant{\blue \delta_1'^{-1}(\text{or}~\bar{\delta}_1^{-1})\geqslant\lambda_p\geqslant\delta_1^{-1}}\geqslant\big(k(p)\sigma_p\big)^{-1},$$
where
$$\aligned&\delta_1=\sup_{i\in E}\bigg[\frac{1}{\hat{\nu}[i, N]^{1/p'}}\sum_{j=i}^N\hat{\nu}_j\bigg(\sum_{k=0}^{j}\mu_k\hat{\nu}[k, N]^{(p-1)/p^*}\bigg)^{p'-1}\bigg]^{p-1},\\
&\delta_1'=\sup_{\ell\in E}\frac{1}{\hat{\nu}[\ell, N]^{p-1}}\bigg[\sum_{j=\ell}^N\hat{\nu}_j\bigg(\sum_{k=0}^{j}\mu_{k}\hat{\nu}[k\vee \ell, N]^{p-1}\bigg)^{p'-1}\bigg]^{p-1},\\
&\bar{\delta}_{1}=\sup_{m\in E}\frac{1}{\hat{\nu}[m,N]}\sum_{j=0}^{N}\mu_{j}\hat{\nu}[j \vee m, N]^p\in[\sigma_p, p\sigma_p].\endaligned$$
{\blue For $1<p\leqslant2$, $\bar\delta_1\leqslant\delta_1'$; for $p\geqslant2$, $\bar\delta_1\geqslant\delta_1'$.}
}
\end{block}

\end{frame}
%----------------------------------------------------------14
\begin{frame}
\frametitle{4. Examples}\small
\begin{block}{} $E=\{0, 1,\cdots, 80\}$. Let $\mu_k=20^{k}$, $\nu_k=20^{k+1}$.
\end{block}
\begin{center}{\includegraphics[width=11.0cm,height=4.5cm]{Hardy-2-7-1.eps}
}\end{center}
Let $p$ vary over $(1.001, 30.001)$
avoiding the singularity at $p=1$.
The curves from top to bottom are
$(k(p) \sigma_p)^{1/p}$, ${\delta_1}^{1/p}$, ${\bar\delta}_1^{1/p}$, ${\delta_1'}^{1/p}$ and $\sigma_p^{1/p}$ res.
The lower bounds  ${\bar\delta}_1^{1/p}$ and ${\delta_1'}^{1/p}$ of $\lambda_p^{-1/p}$ are nearly overlapped.
\end{frame}
%----------------------------------------------------------15
\begin{frame}
\small
\begin{block}{} $E=\{0, 1,\cdots, 40\}$. Let $\mu_k=1$, $\nu_k=1$.
\end{block}
\begin{center}{\includegraphics[width=11.0cm,height=4.5cm]{Hardy-2-7-2.eps}}\end{center}
Let $p$ vary over $(1.0175, 30.0175)$. The curves from top to bottom are
$(k(p)\sigma_p)^{1/p}$, ${\delta_1}^{1/p}$, ${\bar\delta}_1^{1/p}$, ${\delta_1'}^{1/p}$ and $\sigma_p^{1/p}$.
The exact $\lambda_p$ is unknown except that $\lambda_p=\sin^2 \frac{\pi}{2(N+2)}$ when $p=2$.
\end{frame}
%--------------------------------------------------------------------16
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\centerline{ \emph{\textbf{\color{blue} Thank you for your attention!}}}

\end{frame}

\end{CJK*}

\end{document}
