Transition Layer for the Heterogeneous Allen-Cahn Equation

We consider the equation \e^{2}\Delta u=(u-a(x))(u^2-1) in $\Omega$, $\frac{\partial u}{\partial \nu} =0$ on $\partial \Omega$, where $\Omega$ is a smooth and bounded domain in $\R^n$, $\nu$ the outer unit normal to \pa\Omega, and $a$ a smooth function satisfying $-1<a(x)<1$ in \ov{\Omega}. We set $K$, $\Omega_+$ and $\Omega_-$ to be respectively the zero-level set of $a$, {a>0} and {a<0}. Assuming $\nabla a \neq 0$ on $K$ and $a\ne 0$ on $\partial \Omega$, we show that there exists a sequence \e_j \to 0 such that the above equation has a solution u_{\e_j} which converges uniformly to $\pm 1$ on the compact sets of \O_{\pm} as $j \to + \infty$.Comment: 25 page

Discovery of optical candidate supernova remnants in Sagittarius

During an [O III] survey for planetary nebulae, we identified a region in Sagittarius containing several candidate Supernova Remnants and obtained deep optical narrow-band images and spectra to explore their nature. The images of the unstudied area have been obtained in the light of Halpha+[N II], [S II] and [O III]. The resulting mosaic covers an area of 1.4x1.0 deg^2 where filamentary and diffuse emission was discovered, suggesting the existence of more than one supernova remnants (SNRs) in the area. Deep long slit spectra were also taken of eight different regions. Both the flux calibrated images and the spectra show that the emission from the filamentary structures originates from shock-heated gas, while the photo-ionization mechanism is responsible for the diffuse emission. Part of the optical emission is found to be correlated with the radio at 4850 MHz suggesting their association, while the WISE infrared emission found in the area at 12 and 22 micron marginally correlates with the optical. The presence of the [O III] emission line in one of the candidate SNRs suggests shock velocities into the interstellar "clouds" between 120 and 200 km/s, while the absence in the other indicates slower shock velocities. For all candidate remnants the [S II] 6716/6731 ratio indicates electron densities below 240 cm^{-3}, while the Halpha emission has been measured to be between 0.6 to 41x10^{-17} erg/s/cm^2/arcsec^2. The existence of eight pulsars within 1.5deg away from the center of the candidate SNRs also supports the scenario of many SNRs in the area as well as that the detected optical emission could be part of a number of supernovae explosions.Comment: 15 pages, 6 figures, accepted for publication in Astronomy & Astrophysic