\documentclass{beamer}
%\textwidth=9.5in
%\textheight=6in
%\topmargin -.7in
\mode<presentation>{\usetheme{default}\setbeamercovered{invisible}\usecolortheme{crane}}
\usepackage[english]{babel}
\usepackage[latin1]{inputenc}
\usepackage{times}
\usepackage[T1]{fontenc}
\usepackage{xcolor}
\usepackage{amstext,latexsym,amsbsy,amssymb,amsmath,amsthm}
\pagenumbering{arabic}
\newcommand{\hh}{\hspace*{.48in}}
\newcommand{\hk}{\hspace*{.24in}}
\newcommand{\hhh}{\hspace*{.72in}}
\newcommand{\nh}{\hspace*{-.12in}}
\newcommand{\nhh}{\hspace*{-.24in}}
% Định nghĩa các tiêu đề
\renewcommand{\chaptername}{Chương}
\renewcommand{\appendixname}{PHỤ LỤC}
\newcommand{\hty}{\rightharpoonup}
\newcommand{\rong}{\emptyset}
\newtheorem{tm}{ }
\newtheorem{dn}{\large Định nghĩa}[section]
\newtheorem{dly}{\large Định lý}[section]
\newtheorem{chy}{\large Chú ý}[section]
\newtheorem{bd}{\large Bổ đề}[section]
\newtheorem{md}{\large Mệnh đề}[section]
\newtheorem{hq}{\large Hệ quả}[section]
\newtheorem{ly}{\large Lưu ý}[section]
\newtheorem{thm}{Theorem}[section]
\newtheorem{lem}[thm]{Lemma}
\newtheorem{eg}[thm]{Example}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{defn}[thm]{Definition}
\newtheorem{rem}[thm]{Remark}
\newtheorem{ntn}[thm]{Notation}
\newenvironment{prf}{{\noindent \textbf{Proof:}\ }}{\hfill $\Box$\\ \smallskip}
\numberwithin{equation}{section}
% Định nghĩa các công thức toán, các bạn có thể xóa bỏ phần này
\def\inty{\infty}
\def\beqn{\begin{eqnarray}}\def\eeqn{\end{eqnarray}}
\def\bqq{\[}\def\eqq{\]}
\def\lef{\lefteqn}
\def\z{\textstyle }
\newcommand{\da}[3]{#1\zoz{#2}\zuz{#3}}
\newcommand{\ssm}[2]{\displaystyle\sum_{\mbox{\scriptsize$\begin{array}{l}
\mbox{\scriptsize$#1$}\\\mbox{\scriptsize$#2$}\end{array}$}}}
\newcommand{\lf}[2]{\mbox{\large$\frac{#1}{#2}$}}
\newcommand{\Lf}[2]{{\displaystyle\frac{\z #1}{\z #2}}}
\newcommand{\zu}[1]{_{{}_{\z #1}}}
\newcommand{\zo}[1]{^{\z #1}}
\newcommand{\zuz}[1]{_{\z{}_{#1}}}
\newcommand{\zoz}[1]{^{#1}}
\def\NN{\mbox{$I\hspace{-.06in}N$}}
\def\ZZ{\mbox{$Z\hspace{-.08in}Z\hspace{.05in}$}}
\def\QQ{\mbox{$Q\hspace{-.11in}\protect\raisebox{.5ex}{\tiny$/$}
\hspace{.06in}$}}
\def\RR{\mbox{$I\hspace{-.06in}R$}}
\def\CC{\mbox{$C\hspace{-.11in}\protect\raisebox{.5ex}{\tiny$/$}
\hspace{.06in}$}}
\def\HH{\mbox{$I\hspace{-.06in}H$}}
\def\bu{\mbox{$\bullet$}}
\def\jnt{\displaystyle\int}
\def\xo{\vspace{.05in}\\\hspace*{5in}\mbox{\huge$\Box$} }
\newcommand{\ii}[1]{\displaystyle\int_{{}_{\z#1}}}
\newcommand{\ik}[2]{\displaystyle\int_{{}_{\z #1}}^{\z #2}}
\def\io{\displaystyle\int_{{}_{\z \Omega}}}
\def\irn{\ii{\RR^{n}}}
\newcommand{\iii}[1]{\displaystyle\int\hspace{-.16in}-\hspace{-.06in}\zu{{}\zu{\z#1}}}
\def\imr{\iii{B(x,3r)}}
\newcommand{\iR}[1]{\displaystyle\int_{{}_{\z \RR^{#1}}}}
\newcommand{\ijz}[2]{\displaystyle\int_{{}_{ #1}}^{ #2}}
\newcommand{\khog}[4]{#1\zuz{#2}\zoz{#3}(#4)}
\newcommand{\vh}[2]{\langle #1,#2\rangle}
\def\sij{\displaystyle\sum_{i,j=1}^{n}} \def\sj{\displaystyle\sum_{j=1}^{n}}
\newcommand{\su}[2]{\displaystyle\sum\zu{#1}\zo{#2}}
\newcommand{\suz}[2]{\displaystyle\sum\zuz{#1}\zoz{#2}}
\newcommand{\suu}[4]{\displaystyle\sum_{#1}^{#2}\displaystyle\sum_{#3}^{#4}}
\newcommand{\sss}[3]{\displaystyle\sum_{\begin{array}{l}\mbox{\W$#1$}\\
\mbox{\W$#3$}\end{array}}^{#2}}
\newcommand{\ssso}[3]{{\displaystyle\sum\zu{#1}\zo{#2}}\hspace{-.3in}\zu{{}\zu{
{}\zu{{}\zu{{}\zu{{}\zu{\zu{{}_{#3}}}}}}}}\hspace{-.12in}}
\newcommand{\sssss}[5]{{\displaystyle\sum_{#1}^{#2}\displaystyle\sum_{#3}^{#4}}
\hspace{-.5in}_{{}\zu{{}\zu{{}\zu{{}\zu{{}\zu{{}\zu{{}_{#5}}}}}}}}}
\newcommand{\ca}[1]{{\cal#1}}
%Định nghĩa màu cho chữ viết, các bạn không nên xóa bỏ phần này
\newcommand{\doo}[1]{\textcolor{red}{#1}}
\newcommand{\xanh}[1]{\textcolor{green}{#1}}
\newcommand{\duong}[1]{\textcolor{blue}{#1}}
\newcommand{\bich}[1]{\textcolor{cyan}{#1}}
\newcommand{\hong}[1]{\textcolor{magenta}{#1}}
\newcommand{\vang}[1]{\textcolor{yellow}{#1}}
%Định nghĩa màu cho khung và chữ viết, các bạn không nên xóa bỏ phần này
\setbeamercolor{vangxanh}{fg=blue,bg=yellow!20!white}
\setbeamercolor{vangden}{fg=black,bg=yellow!20!white}
\setbeamercolor{vangdo}{fg=red,bg=yellow!20!white}
\setbeamercolor{camden}{fg=black,bg=orange!20!white}
\setbeamercolor{camxanh}{fg=blue,bg=orange!20!white}
\setbeamercolor{bichdo}{fg=red,bg=cyan!15!white}
\setbeamercolor{bichden}{fg=black,bg=cyan!15!white}
\setbeamercolor{hongxanh}{fg=blue,bg=magenta!15!white}
\setbeamercolor{hongden}{fg=black,bg=magenta!15!white}
\setbeamercolor{lado}{fg=red,bg=lime!40}
\setbeamercolor{laden}{fg=black,bg=lime!40}
\setbeamercolor{laxanh}{fg=blue,bg=lime!40}
\newcommand{\bm}[2]{\begin{beamercolorbox}[sep=.1em,wd=4.8in]{#1}
#2
\end{beamercolorbox}}
% Định nghĩa một số ký hiệu toán, các bạn có thể xóa bỏ phần này
\newcommand{\id}{{\rm id}}
\newcommand{\ti}{\tilde}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}
\newcommand{\cb}{{\rm cb}}
\newcommand{\CB}{{\rm CB}}
\newcommand{\abs}[1]{\left\vert#1\right\vert}
\newcommand{\im}{{\rm Im\ \!}}
\newcommand{\sph}{\mathfrak{S}_1}
\newcommand{\CS}{\mathcal{S}}
\newcommand{\CK}{\mathcal{K}}
\newcommand{\CL}{\mathcal{L}}
\newcommand{\OX}{\mathbf{X}}
\newcommand{\OY}{\mathbf{Y}}
\newcommand{\OZ}{\mathbf{Z}}
\newcommand{\BK}{\mathbf{K}}
\newcommand{\QS}{\mathcal{Q}}
\newcommand{\PS}{\mathcal{P}}
\newcommand{\QM}{{\rm qM}}
\newcommand{\PM}{{\rm pMor}}
\newcommand{\M}{{\rm Mor}}
\newcommand{\st}{{\empty^\ast}}
\newcommand{\reg}{{\rm reg}}
\newcommand{\bl}{\left}
\newcommand{\br}{\right}
\newcommand{\es}{{\rm es}}
\newcommand{\bal}{\mathfrak{B}_1}
\newcommand{\BN}{\mathbb{N}}
\begin{document}
\begin{frame}
\bm{bichdo}{\begin{center}{\Large \bf GLOBAL EIGENVALUE-CROSSING AND MULTIPLICITY OF SOLUTIONS TO ELLIPTIC EQUATIONS}\end{center}}
\medskip
\bm{camden}{\begin{center}{Duong Minh Duc \\
Vietnam National University at Hochiminh City (Vietnam)\\
\doo{Nguyen Hoang Loc\\
Department of Mathematics, University of Utah (USA)}\\
\duong{Nguyen Le Luc\\
Mathematical Institute, Oxford University (UK)}\\
\bich{Le Quang Nam\\
Department of Mathematics, Columbia University, New York (USA)}\\
\hong{Nguyen Trung Tuyen\\
Department of Mathematics, Indiana University, Indiana (USA)}}\end{center}}
\bm{hongxanh}{\begin{center}{DIFFERENTIAL AND DIFFERENCE EQUATIONS \\
Hanoi and Halong, 29-31 October, 2009}\end{center}}
\end{frame}
\section{}
\begin{frame}
\bm{hongxanh}{Let $\Omega$ be a smooth bounded open subset of $\RR^N~(N\geq 3)$, and $(W^{1,2}_0(\Omega ),\|.\|)$ be the usual Sobolev space. Let $f$ be a continuously differentiable on $\Omega\times \RR$. In this paper, we study the multiplicity of weak solutions in $(W^{1,2}_0(\Omega )$ to the following boundary value problem:
\begin{equation}
\left \{ \begin{array}{ll}\Delta u+ f(x,u)=0&\mbox{ in }\Omega \\\hhh ~~u=0&\mbox{ on }\partial \Omega\end{array}\right.
\label{eqn01}
\end{equation}\hh}
\medskip
\pause
\bm{bichden}{Let $\lambda _1<\lambda _2\leq \ldots $ be the eigenvalues of the Laplacian operator $-\Delta$ with homogeneous Dirichlet boundary condition on $\Omega$, and let $\varphi _1,~\varphi _2,\ldots $ be their corresponding eigenfunctions in $W^{1,2}_0(\Omega )$. \pause \doo{If $\Lf{f(x,u)}{u}$ crosses some $\lambda_{i}$ when $|u|$ varies from $0$ to $\infty$ uniformly on $\Omega$, the problem has been extensively studied by many mathematicians.} \pause \hong{If $\Lf{f(x,u)}{u}$ does not cross any $\lambda_{i}$, the problem $(\ref{eqn01})$ may not have any solution.} \pause However, in the double-resonance case, Zou obtained multiplicity results when $\Lf{f(x,u)}{u}$ does not cross any $\lambda_{i}$. }
\end{frame}
\begin{frame}
\bm{camxanh}{Let $\mu$ and $\nu$ be real numbers such that $\mu < \lambda_{m}$ and $\lambda_{k} < \nu$. We have an eigenvalue-crossing from $\mu$ to $\nu$. We explain this eigenvalue-crossing in global sense as follows.}
\medskip
\pause
\bm{hongden}{Denote by $Y$ the subspace of $W^{1,2}_0(\Omega )$ spanned by $\{\varphi _1,\varphi _2,\ldots,\varphi _k\}$ and by $Z$ the orthogonal complement of $Y$ in $W^{1,2}_0(\Omega )$. Note that
\begin{eqnarray}
\int _{\Omega}|\nabla \varphi_{j}|^{2}dx &=& \lambda_{j} \int _{\Omega}|\varphi_{j}|^{2}dx
\hhh\forall j\in \{1,2,\cdots\},\\
\int _{\Omega}|\nabla(\sum_{j=1}^{\infty}\alpha_{j} \varphi_{j})|^{2}dx &=& \sum_{j=1}^{\infty}\lambda_{j}|\alpha_{j}|^{2} \int _{\Omega}|\varphi_{j}|^{2}dx.\label{01}
\end{eqnarray} }
\medskip
\pause
\bm{bichden}{Put $c_{1} = \min\{\lambda_{1}-\mu,\nu-\lambda_{k}\}$, by $(\ref{01})$ we have
\begin{eqnarray}
\int _{\Omega}\left (|\nabla z|^2-\mu z^2 \right)dx&\ge& c_1 \|z\|^2 \hh\forall z\in Z,\\
\int _{\Omega}\left(|\nabla y|^2-\nu y^2 \right)dx&\le& -(\nu-\lambda_{k})\|y\|^2 \hh\forall ~y \in Y,
\end{eqnarray}}
\end{frame}
\end{document}