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

Gas/liquid flow behaviours in a downward section of large diameter vertical serpentine pipes

An experimental study on air/water flow behaviours in a 101.6 mm i.d. vertical pipe with a serpentine configuration is presented. The experiments are conducted for superficial gas and liquid velocities ranging from 0.15 to 30 m/s and 0.07 to 1.5 m/s, respectively. The bend effects on the flow behaviours are significantly reduced when the flow reaches an axial distance of 30 pipe diameters or more from the upstream bend. The mean film thickness data from this study has been used to compare with the predicted data using several falling film correlations and theoretical models. It was observed that the large pipe data exhibits different tendencies and this manifests in the difference in slope when the dimensionless film thickness is plotted as a power law function of the liquid film Reynolds number

Interfacial shear in adiabatic downward gas/liquid co-current annular flow in pipes

Interfacial friction is one of the key variables for predicting annular two-phase flow behaviours in vertical pipes. In order to develop an improved correlation for interfacial friction factor in downward co-current annular flow, the pressure gradient, film thickness and film velocity data were generated from experiments carried out on Cranfield University’s Serpent Rig, an air/water two-phase vertical flow loop of 101.6 mm internal diameter. The air and water superficial velocity ranges used are 1.42–28.87 and 0.1–1.0 m/s respectively. These correspond to Reynolds number values of 8400–187,000 and 11,000–113,000 respectively. The correlation takes into account the effect of pipe diameter by using the interfacial shear data together with dimensionless liquid film thicknesses related to different pipe sizes ranging from 10 to 101.6 mm, including those from published sources by numerous investigators. It is shown that the predictions of this new correlation outperform those from previously reported studies

A Multivariate Model of Strategic Asset Allocation

Much recent work has documented evidence for predictability of asset returns. We show how such predictability can affect the portfolio choices of long-lived investors who value wealth not for its own sake but for the consumption their wealth can support. We develop an approximate solution method for the optimal consumption and portfolio choice problem of an infinitely-lived investor with Epstein-Zin utility who faces a set of asset returns described by a vector autoregression in returns and state variables. Empirical estimates in long-run annual and postwar quarterly US data suggest that the predictability of stock returns greatly increases the optimal demand for stocks. The role of nominal bonds in long-term portfolios depends on the importance of real interest rate risk relative to other sources of risk. We extend the analysis to consider long-term inflation-indexed bonds and find that these bonds greatly increase the utility of conservative investors, who should hold large positions when they are available.