Concavity properties for free boundary elliptic problems

We prove some concavity properties connected to nonlinear Bernoulli type free boundary problems. In particular, we prove a Brunn-Minkowski inequality and an Urysohn's type inequality for the Bernoulli Constant and we study the behaviour of the free boundary with respect to the given boundary data. Moreover we prove a uniqueness result regarding the interior non-linear Bernoulli problem.Comment: 12 pages. This is a revised version of the already published paper, which includes the corrections contained in the Corrigendum available online at http://dx.doi.org/10.1016/j.na.2009.11.03

Rotating Globular Clusters

Internal rotation is considered to play a major role in the dynamics of some globular clusters. However, in only few cases it has been studied by quantitative application of realistic and physically justified global models. Here we present a dynamical analysis of the photometry and three-dimensional kinematics of omega Cen, 47 Tuc, and M15, by means of a recently introduced family of self-consistent axisymmetric rotating models. The three clusters, characterized by different relaxation conditions, show evidence of differential rotation and deviations from sphericity. The combination of line-of-sight velocities and proper motions allows us to determine their internal dynamics, predict their morphology, and estimate their dynamical distance. The well-relaxed cluster 47 Tuc is very well interpreted by our model; internal rotation is found to explain the observed morphology. For M15, we provide a global model in good agreement with the data, including the central behavior of the rotation profile and the shape of the ellipticity profile. For the partially relaxed cluster omega Cen, the selected model reproduces the complex three-dimensional kinematics; in particular the observed anisotropy profile, characterized by a transition from isotropy, to weakly-radial anisotropy, and then to tangential anisotropy in the outer parts. The discrepancy found for the steep central gradient in the observed line-of-sight velocity dispersion profile and for the ellipticity profile is ascribed to the condition of only partial relaxation of this cluster and the interplay between rotation and radial anisotropy.Comment: 19 pages, 14 figures, accepted for publication in the Astrophysical Journa

Kinematic fingerprint of core-collapsed globular clusters

Dynamical evolution drives globular clusters toward core collapse, which strongly shapes their internal properties. Diagnostics of core collapse have so far been based on photometry only, namely on the study of the concentration of the density profiles. Here we present a new method to robustly identify core-collapsed clusters based on the study of their stellar kinematics. We introduce the \textit{kinematic concentration} parameter, $c_k$, the ratio between the global and local degree of energy equipartition reached by a cluster, and show through extensive direct $N$-body simulations that clusters approaching core collapse and in the post-core collapse phase are strictly characterized by $c_k>1$. The kinematic concentration provides a suitable diagnostic to identify core-collapsed clusters, independent from any other previous methods based on photometry. We also explore the effects of incomplete radial and stellar mass coverage on the calculation of $c_k$ and find that our method can be applied to state-of-art kinematic datasets.Comment: Accepted for publication in MNRAS Lette

SBV Regularity for Genuinely Nonlinear, Strictly Hyperbolic Systems of Conservation Laws in one space dimension

We prove that if $t \mapsto u(t) \in \mathrm {BV}(\R)$ is the entropy solution to a $N \times N$ strictly hyperbolic system of conservation laws with genuinely nonlinear characteristic fields $$u_t + f(u)_x = 0,$$ then up to a countable set of times $\{t_n\}_{n \in \mathbb N}$ the function $u(t)$ is in $\mathrm {SBV}$, i.e. its distributional derivative $u_x$ is a measure with no Cantorian part. The proof is based on the decomposition of $u_x(t)$ into waves belonging to the characteristic families $$u(t) = \sum_{i=1}^N v_i(t) \tilde r_i(t), \quad v_i(t) \in \mathcal M(\R), \ \tilde r_i(t) \in \mathrm R^N,$$ and the balance of the continuous/jump part of the measures $v_i$ in regions bounded by characteristics. To this aim, a new interaction measure \mu_{i,\jump} is introduced, controlling the creation of atoms in the measure $v_i(t)$. The main argument of the proof is that for all $t$ where the Cantorian part of $v_i$ is not 0, either the Glimm functional has a downward jump, or there is a cancellation of waves or the measure $\mu_{i,\mathrm{jump}}$ is positive

Global Structure of Admissible BV Solutions to Piecewise Genuinely Nonlinear, Strictly Hyperbolic Conservation Laws in One Space Dimension

The paper describes the qualitative structure of an admissible BV solution to a strictly hyperbolic system of conservation laws whose characteristic families are piecewise genuinely nonlinear. More precisely, we prove that there are a countable set of points \u398 and a countable family of Lipschitz curves T{script} such that outside T{script} 2a \u398 the solution is continuous, and for all points in T{script}{set minus}\u398 the solution has left and right limit. This extends the corresponding structural result in [7] for genuinely nonlinear systems. An application of this result is the stability of the wave structure of solution w.r.t. -convergence. The proof is based on the introduction of subdiscontinuities of a shock, whose behavior is qualitatively analogous to the discontinuities of the solution to genuinely nonlinear systems

V4743 Sgr, a magnetic nova?

Two XMM Newton observations of Nova V4743 Sgr (Nova Sgr 2002) were performed shortly after it returned to quiescence, 2 and 3.5 years after the explosion. The X-ray light curves revealed a modulation with a frequency of ~0.75 mHz, indicating that V4743 Sgr is most probably an intermediate polar (IP). The X-ray spectra have characteristics in common with known IPs, with a hard thermal plasma component that can be fitted only assuming a partially covering absorber. In 2004 the X-ray spectrum had also a supersoft blackbody-like component, whose temperature was close to that of the white dwarf (WD) in the supersoft X-ray phase following the outburst, but with flux by at least two orders of magnitude lower. In quiescent IPs, a soft X-ray flux component originates at times in the polar regions irradiated by an accretion column, but the supersoft component of V4743 Sgr disappeared in 2006, indicating a possible origin different from accretion. We suggest that it may have been due to an atmospheric temperature gradient on the WD surface, or to continuing localized thermonuclear burning at the bottom of the envelope, before complete turn-off. An optical spectrum obtained with SALT 11.5 years after the outburst showed a prominent He II 4686A line and the Bowen blend, which reveal a very hot region, but with peak temperature shifted to the ultraviolet (UV) range. V4743 Sgr is the third post-outburst nova and IP candidate showing a low-luminosity supersoft component in the X-ray flux a few years after the outburst.Comment: 9 pages, 5 figures, accepted to MNRA

A uniqueness result for the decomposition of vector fields in Rd

Given a vector field \u3c1(1,b) 08Lloc1(R+ 7Rd,Rd+1) such that divt,x(\u3c1(1,b)) is a measure, we consider the problem of uniqueness of the representation \u3b7 of \u3c1(1 , b) Ld+1 as a superposition of characteristics \u3b3:(t\u3b3-,t\u3b3+)\u2192Rd, \u3b3\u2d9 (t) = b(t, \u3b3(t)). We give conditions in terms of a local structure of the representation \u3b7 on suitable sets in order to prove that there is a partition of Rd+1 into disjoint trajectories \u2118a, a 08 A, such that the PDE divt,x(u\u3c1(1,b)) 08M(Rd+1),u 08L 1e(R+ 7Rd),can be disintegrated into a family of ODEs along \u2118a with measure r.h.s. The decomposition \u2118a is essentially unique. We finally show that b 08Lt1(BVx)loc satisfies this local structural assumption and this yields, in particular, the renormalization property for nearly incompressible BV vector fields

FORWARD UNTANGLING AND APPLICATIONS TO THE UNIQUENESS PROBLEM FOR THE CONTINUITY EQUATION

We introduce the notion of forward untangled Lagrangian representation of a measure-divergence vector-measure rho(1, b), where rho is an element of M+(Rd+1) and b : Rd+1 -> R-d is a rho-integrable vector field with div(t,x)(rho(1, b)) = mu is an element of M(R x R-d): forward untangling formalizes the notion of forward uniqueness in the language of Lagrangian representations. We identify local conditions for a Lagrangian representation to be forward untangled, and we show how to derive global forward untangling from such local assumptions. We then show how to reduce the PDE div(t,x)(rho(1, b)) = mu on a partition of R+ x R-d obtained concatenating the curves seen by the Lagrangian representation. As an application, we recover known well posedeness results for the flow of monotone vector fields and for the associated continuity equation