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

Large gauge invariant non-standard neutrino interactions

Theories beyond the Standard Model must necessarily respect its gauge symmetry. This implies strict constraints on the possible models of non-standard neutrino interactions, which we analyze. The focus is set on the effective low-energy dimension six and eight operators involving four leptons, decomposing them according to all possible tree-level mediators, as a guide for model building. The new couplings are required to have sizeable strength, while processes involving four charged leptons are required to be suppressed. For non-standard interactions in matter, only diagonal tau-neutrino interactions can escape these requirements and can be allowed to result from dimension six operators. Large non-standard neutrino interactions from dimension eight operators alone are phenomenologically allowed in all flavour channels and shown to require at least two new mediator particles. The new couplings must obey general cancellation conditions both at the dimension six and eight levels, which result from expressing the operators obtained from the mediator analysis in terms of a complete basis of operators. We illustrate with one example how to apply this information to model building.Comment: 34 pages, 4 figures, 5 tables. Final version in PR

Optical and X-ray Properties of the Swift BAT-detected AGN

The Swift Gamma-Ray Burst satellite has detected a largely unbiased towards absorption sample of local ($\approx 0.03$) AGN, based solely on their 14--195 keV flux. In the first 9 months of the survey, 153 AGN sources were detected. The X-ray properties in the 0.3--10 keV band have been compiled and presented based on analyses with XMM-Newton, Chandra, Suzaku, and the Swift XRT (Winter et al. 2009). Additionally, we have compiled a sub-sample of sources with medium resolution optical ground-based spectra from the SDSS or our own observations at KPNO. In this sample of 60 sources, we have classified the sources using standard emission line diagnostic plots, obtained masses for the broad line sources through measurement of the broad H$\beta$ emission line, and measured the [OIII] 5007\AA luminosity of this sample. Based on continuum fits to the intrinsic absorption features, we have obtained clues about the stellar populations of the host galaxies. We now present the highlights of our X-ray and optical studies of this unique sample of local AGNs, including a comparison of the 2--10 keV and 14--195 keV X-ray luminosities with the [OIII] 5007\AA luminosity and the implications of our results towards measurements of bolometric luminosities.Comment: 4 pages, 2 figures, to appear in proceedings for 'X-ray Astronomy 2009', Bologna 09/2009, AIP Conference Series, Eds. A. Comastri, M. Cappi, L. Angelin