Uniqueness of the blow-up at isolated singularities for the Alt-Caffarelli functional

In this paper we prove uniqueness of blow-ups and $C^{1,\log}$-regularity for the free-boundary of minimizers of the Alt-Caffarelli functional at points where one blow-up has an isolated singularity. We do this by establishing a (log-)epiperimetric inequality for the Weiss energy for traces close to that of a cone with isolated singularity, whose free-boundary is graphical and smooth over that of the cone in the sphere. With additional assumptions on the cone, we can prove a classical epiperimetric inequality which can be applied to deduce a $C^{1,\alpha}$ regularity result. We also show that these additional assumptions are satisfied by the De Silva-Jerison-type cones, which are the only known examples of minimizing cones with isolated singularity. Our approach draws a connection between epiperimetric inequalities and the \L ojasiewicz inequality, and, to our knowledge, provides the first regularity result at singular points in the one-phase Bernoulli problem.Comment: 37 pages. To appear in Duke Math Journa

(Log-)epiperimetric inequality and regularity over smooth cones for almost Area-Minimizing currents

We prove a new logarithmic epiperimetric inequality for multiplicity-one stationary cones with isolated singularity by flowing in the radial direction any given trace along appropriately chosen directions. In contrast to previous epiperimetric inequalities for minimal surfaces (e.g. those of Reifenberg, Taylor and White), we need no a priori assumptions on the structure of the cone (e.g. integrability). Moreover, if the cone is integrable (not only through rotations), we recover the classical epiperimetric inequality. As a consequence we deduce a new $\varepsilon$-regularity result for almost area-minimizing currents at singular points, where at least one blow-up is a multiplicity-one cone with isolated singularity. This result is similar to the one for stationary varifolds of Leon Simon, but independent from it since almost minimizers do not satisfy any equation

JSTAR: Jon McBride Software Testing & Research Lab

No abstract availabl

Early-type galaxies at large galactocentric radii - I. Stellar kinematics and photometric properties

We present the results of a combined analysis of the kinematic and photometric properties at large galactocentric radii of a sample of 14 low-luminosity early-type galaxies in the Fornax and Virgo clusters. From Gemini South GMOS long-slit spectroscopic data we measure radial profiles of the kinematic parameters v_{rot}, sigma, h_{3}, and h_{4} out to ~ 1 - 3 effective radii. Multi-band imaging data from the HST/ACS are employed to evaluate surface brightness profiles and isophotal shape parameters of ellipticity, position angle and discyness/boxiness. The galaxies are found to host a cold and old stellar component which extend to the largest observed radii and that is the dominant source of their dynamical support. The prevalence of discy-shaped isophotes and the radial variation of their ellipticity are signatures of a gradual gas dissipation. An early star-forming collapse appears to be the main mechanism acting in the formation of these objects. Major mergers are unlikely to have occurred in these galaxies. We can not rule out a minor merging origin for these galaxies, but a comparison of our results with model predictions of different merger categories places some constraints on the possible merger progenitors. These merger events are required to happen at high-redshift (i.e., z > 1), between progenitors of different mass ratio (at least 3:1) and containing a significant amount of gas (i.e., > 10 percent). A further scenario is that the low-luminosity galaxies were originally late-type galaxies, whose star formation has been truncated by removal of gas and subsequently the disc has been dynamically heated by high speed encounters in the cluster environment.Comment: 19 pages, 16 figures (Contact author for high resolution version of figures 1,2,3), MNRAS, accepted. The second paper of the series "Early-type galaxies at large galactocentric radii - II. Metallicity gradients and the [Z/H]-mass, [alpha/Fe] mass relations" can be found at arXiv:1006.169

The mass-metallicity gradient relation of early-type galaxies

We present a newly observed relation between galaxy mass and radial metallicity gradients of early-type galaxies. Our sample of 51 early-type galaxies encompasses a comprehensive mass range from dwarf to brightest cluster galaxies. The metallicity gradients are measured out to one effective radius by comparing nearly all of the Lick absorption-line indices to recent models of single stellar populations. The relation shows very different behaviour at low and high masses, with a sharp transition being seen at a mass of ~ 3.5 x 10^10 M_sun (velocity dispersion of ~140 km/s, M_B ~ -19). Low-mass galaxies form a tight relation with mass, such that metallicity gradients become shallower with decreasing mass and positive at the very low-mass end. Above the mass transition point several massive galaxies have steeper gradients, but a clear downturn is visible marked by a broad scatter. The results are interpreted in comparison with competing model predictions. We find that an early star-forming collapse could have acted as the main mechanism for the formation of low-mass galaxies, with star formation efficiency increasing with galactic mass. The high-mass downturn could be a consequence of merging and the observed larger scatter a natural result of different merger properties. These results suggest that galaxies above the mass threshold of ~ 3.5 x 10^10 M_sun might have formed initially by mergers of gas-rich disc galaxies and then subsequently evolved via dry merger events. The varying efficiency of the dissipative merger-induced starburst and feedback processes have shaped the radial metallicity gradients in these high-mass systems.Comment: 5 pageg, 3 figures, accepted by ApJ Lette

Metallicity Gradients at Large Galactocentric Radii Using the Near-infrared Calcium Triplet

We describe a new spectroscopic technique for measuring radial metallicity gradients out to large galactocentric radii. We use the DEIMOS multi-object spectrograph on the Keck telescope and the galaxy spectrum extraction technique of Proctor et al. (2009). We also make use of the metallicity sensitive near-infrared (NIR) Calcium triplet (CaT) features together with single stellar population models to obtain metallicities. Our technique is applied as a pilot study to a sample of three relatively nearby (<30 Mpc) intermediate-mass to massive early-type galaxies. Results are compared with previous literature inner region values and generally show good agreement. We also include a comparison with profiles from dissipational disk-disk major merger simulations. Based on our new extended metallicity gradients combined with other observational evidence and theoretical predictions, we discuss possible formation scenarios for the galaxies in our sample. The limitations of our new technique are also discussed.Comment: 13 Pages, 9 Figures, 7 Tables, Accepted for publication in MNRA

Probing the 2-D kinematic structure of early-type galaxies out to 3 effective radii

We detail an innovative new technique for measuring the 2-D velocity moments (rotation velocity, velocity dispersion and Gauss-Hermite coefficients h$_3$ and h$_4$) of the stellar populations of galaxy halos using spectra from Keck DEIMOS multi-object spectroscopic observations. The data are used to reconstruct 2-D rotation velocity maps. Here we present data for five nearby early-type galaxies to ~3 effective radii. We provide significant insights into the global kinematic structure of these galaxies, and challenge the accepted morphological classification in several cases. We show that between 1-3 effective radii the velocity dispersion declines very slowly, if at all, in all five galaxies. For the two galaxies with velocity dispersion profiles available from planetary nebulae data we find very good agreement with our stellar profiles. We find a variety of rotation profiles beyond 1 effective radius, i.e rotation speed remaining constant, decreasing \emph{and} increasing with radius. These results are of particular importance to studies which attempt to classify galaxies by their kinematic structure within one effective radius, such as the recent definition of fast- and slow- rotator classes by the SAURON project. Our data suggests that the rotator class may change when larger galacto-centric radii are probed. This has important implications for dynamical modeling of early-type galaxies. The data from this study are available on-line.Comment: 20 pages, 22 figures, Accepted for publication in MNRA