A Higher-Order Energy Expansion to Two-Dimensional Singularly Neumann Problems

Of concern is the following singularly perturbed semilinear elliptic problem \begin{equation*} \left\{ \begin{array}{c} \mbox{${\epsilon}^2\Delta u -u+u^p =0$ in $\Omega$}\\ \mbox{$u>0$ in $\Omega$ and $\frac{\partial u}{\partial \nu}=0$ on $\partial \Omega$}, \end{array} \right. \end{equation*} where $\Omega$ is a bounded domain in ${\mathbf{R}}^N$ with smooth boundary $\partial \Omega$, $\epsilon>0$ is a small constant and $1< p<\left(\frac{N+2}{N-2}\right)_+$. Associated with the above problem is the energy functional $J_{\epsilon}$ defined by \begin{equation*} J_{\epsilon}[u]:=\int_{\Omega}\left(\frac{\epsilon^2}{2}{|\nabla u|}^2 +\frac{1}{2}u^2 -F(u)\right)dx \end{equation*} for $u\in H^1(\Omega)$, where $F(u)=\int_{0}^{u}s^p ds$. Ni and Takagi (\cite{nt1}, \cite{nt2}) proved that for a single boundary spike solution $u_{\epsilon}$, the following asymptotic expansion holds: \begin{equation*} (1) \ \ \ \ \ \ \ \ J_{\epsilon}[u_{\epsilon}]=\epsilon^{N} \left[\frac{1}{2}I[w]-c_1 \epsilon H(P_{\epsilon})+o(\epsilon)\right], \end{equation*} where $I[w]$ is the energy of the ground state, $c_1 >0$ is a generic constant, $P_{\epsilon}$ is the unique local maximum point of $u_{\epsilon}$ and $H(P_{\epsilon})$ is the boundary mean curvature function at $P_{\epsilon}\in \partial \Omega$. Later, Wei and Winter (\cite{ww3}, \cite{ww4}) improved the result and obtained a higher-order expansion of $J_{\epsilon}[u_{\epsilon}]$: \begin{equation*} (2) \ \ \ \ \ \ J_{\epsilon}[u_{\epsilon}]=\epsilon^{N} \left[\frac{1}{2}I[\omega]-c_{1} \epsilon H(P_{\epsilon})+\epsilon^2 [c_2(H(P_\epsilon))^2 +c_{3} R(P_\epsilon)]+o(\epsilon^2)\right], \end{equation*} where $c_2$ and $c_3>0$ are generic constants and $R(P_\epsilon)$ is the scalar curvature at $P_\epsilon$. However, if $N=2$, the scalar curvature is always zero. The expansion (2) is no longer sufficient to distinguish spike locations with same mean curvature. In this paper, we consider this case and assume that $2 \leq p <+\infty$. Without loss of generality, we may assume that the boundary near P\in\partial\Om is represented by the graph $\{ x_2 = \rho_{P} (x_1) \}$. Then we have the following higher order expansion of $J_\epsilon[u_\epsilon]:$ \begin{equation*} (3) \ \ \ \ \ J_\epsilon [u_\epsilon] =\epsilon^N \left[\frac{1}{2}I[w]-c_1 \epsilon H({P_\epsilon})+c_2 \epsilon^2(H({P_\epsilon}))^2 ] +\epsilon^3 [P(H({P_\epsilon}))+c_3S({P_\epsilon})]+o(\epsilon^3)\right], \end{equation*} where H(P_\ep)= \rho_{P_\ep}^{''} (0) is the curvature, $P(t)=A_1 t+A_2 t^2+A_3 t^3$ is a polynomial, $c_1$, $c_2$, $c_3$ and $A_1$, $A_2$,$A_3$ are generic real constants and S(P_\epsilon)= \rho_{P_\ep}^{(4)} (0). In particular $c_3<0$. Some applications of this expansion are given

Measuring the mode volume of plasmonic nanocavities using coupled optical emitters

Metallic optical systems can confine light to deep sub-wavelength dimensions, but verifying the level of confinement at these length scales typically requires specialized techniques and equipment for probing the near-field of the structure. We experimentally measured the confinement of a metal-based optical cavity by using the cavity modes themselves as a sensitive probe of the cavity characteristics. By perturbing the cavity modes with conformal dielectric layers of sub-nm thickness using atomic layer deposition, we find the exponential decay length of the modes to be less than 5% of the free-space wavelength (\lambda) and the mode volume to be of order \lambda^3/1000. These results provide experimental confirmation of the deep sub-wavelength confinement capabilities of metal-based optical cavities.Comment: 11 pages, 4 figure

Livestock products in the Third World: past trends and projections to 1990 and 2000

Meat industry and trade Developing countries Statistics., Dairy products industry Developing countries Statistics., Meat industry and trade Developing countries Forecasting Statistical methods., Dairy products industry Developing countries Forecasting Statistical methods.,

Transnational Strategy of Australian Service Firms: Resource Management for Foreign Expansion.

Australian firms expanding overseas must have a business strategy to exploit any competitive advantage, strongly supported by corporate strategy and management. The focus of this research is on the transnational strategy and management of Australian firms. This study investigates the nature of the integration of home-base and offshore operations of Australian firms to achieve competitive positions in national and world markets. Its purpose is to analyze and assess the impact of the size of the firm, its management and the development of strategies. Conclusions of the study support the fact that management mentality and behavior are important. The study also analyzes management mentality in the context of intra-firm competition or collaboration for resources, in a manner which is consistent with firm's performance. Specifically it demonstrates that management can rationally have different views about priorities for the allocation of resources and international operations yet compete actively for the same resources, or collaborate whenever possible, in order to achieve their own goals

A two-step approach to restorable dynamic QoS routing

Aiming at minimizing the combined bandwidth cost of a pair of disjoint active and backup paths, a popular approach to designing Restorable Dynamic QoS Routing schemes is based on Integer Linear Programming (ILP) formulation. Owing to the very different natures of active and backup paths, we found this approach problematic. In this paper, we propose a simple alternative approach, called two-step routing. In the first step, active path is found using the widest-shortest path (WSP) routing. In the second step, the corresponding backup path is determined using one of the three variants of shortest-widest path (SWP) routing, Basic-SWP, Approximate-SWP and Composite-SWP. Combining both steps, three novel restorable routing algorithms, SBW, SAW and SCW, are obtained. Comparing with the existing best-known algorithms, we show that our two-step routing approach yields noticeably lower call blocking probability, shorter active path length, and adjustable backup path length (depending on the SWP variant adopted). Besides, our two-step routing approach gives a much shorter running time than the ILP approach, which makes it more attractive for dynamic routing.published_or_final_versio

Coarsening Dynamics of a One-Dimensional Driven Cahn-Hilliard System

We study the one-dimensional Cahn-Hilliard equation with an additional driving term representing, say, the effect of gravity. We find that the driving field $E$ has an asymmetric effect on the solution for a single stationary domain wall (or `kink'), the direction of the field determining whether the analytic solutions found by Leung [J.Stat.Phys.{\bf 61}, 345 (1990)] are unique. The dynamics of a kink-antikink pair (`bubble') is then studied. The behaviour of a bubble is dependent on the relative sizes of a characteristic length scale $E^{-1}$, where $E$ is the driving field, and the separation, $L$, of the interfaces. For $EL \gg 1$ the velocities of the interfaces are negligible, while in the opposite limit a travelling-wave solution is found with a velocity $v \propto E/L$. For this latter case ($EL \ll 1$) a set of reduced equations, describing the evolution of the domain lengths, is obtained for a system with a large number of interfaces, and implies a characteristic length scale growing as $(Et)^{1/2}$. Numerical results for the domain-size distribution and structure factor confirm this behavior, and show that the system exhibits dynamical scaling from very early times.Comment: 20 pages, revtex, 10 figures, submitted to Phys. Rev.

Efficient path protection using Bi-directional WDM transmission technology

Bi-directional WDM transmission is a technique that allows wavelengths to be transmitted simultaneously in both directions in a single fiber. Compared with unidirectional WDM systems, it not only saves the cost of deploying extra fibers, but also allows more flexible bandwidth provisioning. To exploit the advantages brought by this flexibility, we investigate path protection based on bi-directional WDM transmission system in this paper. With path protection, a call is accepted if and only if an active data path together with a disjointed backup path can be found in the network. With bi-directional WDM, backup resources sharing in both directions of a fiber is possible. To encourage resources sharing, new cost functions are judiciously designed. Based on them, two original path protection schemes are proposed in this paper, BiPro and BiProLP, where BiProLP aims at further economizing the hardware cost incurred by BiPro. In contrast to the traditional unidirectional schemes, we show that both BiPro and BiProLP can yield noticeably lower call blocking probability, higher system capacity and snorter active/backup path length. © 2005 IEEE.published_or_final_versio