Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces

This paper is motivated by the characterization of the optimal symmetry breaking region in Caffarelli-Kohn-Nirenberg inequalities. As a consequence, optimal functions and sharp constants are computed in the symmetry region. The result solves a longstanding conjecture on the optimal symmetry range. As a byproduct of our method we obtain sharp estimates for the principal eigenvalue of Schr\"odinger operators on some non-flat non-compact manifolds, which to the best of our knowledge are new. The method relies on generalized entropy functionals for nonlinear diffusion equations. It opens a new area of research for approaches related to carr\'e du champ methods on non-compact manifolds. However key estimates depend as much on curvature properties as on purely nonlinear effects. The method is well adapted to functional inequalities involving simple weights and also applies to general cylinders. Beyond results on symmetry and symmetry breaking, and on optimal constants in functional inequalities, rigidity theorems for nonlinear elliptic equations can be deduced in rather general settings.Comment: 33 pages, 1 figur

Relativistic hydrogenic atoms in strong magnetic fields

In the Dirac operator framework we characterize and estimate the ground state energy of relativistic hydrogenic atoms in a constant magnetic field and describe the asymptotic regime corresponding to a large field strength using relativistic Landau levels. We also define and estimate a critical magnetic field beyond which stability is lost

Characterization of the critical magnetic field in the Dirac-Coulomb equation

We consider a relativistic hydrogenic atom in a strong magnetic field. The ground state level depends on the strength of the magnetic field and reaches the lower end of the spectral gap of the Dirac-Coulomb operator for a certain critical value, the critical magnetic field. We also define a critical magnetic field in a Landau level ansatz. In both cases, when the charge Z of the nucleus is not too small, these critical magnetic fields are huge when measured in Tesla, but not so big when the equation is written in dimensionless form. When computed in the Landau level ansatz, orders of magnitude of the critical field are correct, as well as the dependence in Z. The computed value is however significantly too big for a large Z, and the wave function is not well approximated. Hence, accurate numerical computations involving the Dirac equation cannot systematically rely on the Landau level ansatz. Our approach is based on a scaling property. The critical magnetic field is characterized in terms of an equivalent eigenvalue problem. This is our main analytical result, and also the starting point of our numerical scheme

Global impact of the Antarctic ozone hole: Simulations with a 3-D chemical transport model

A study of the Antarctic ozone hole was made with a 3-D chemical transport model using linearized photochemistry for ozone based on observed distribution. The tracer model uses the winds and convection from the GISS general circulation model (8 deg x 10 deg x 23 layers). A 3-year control run of the ozone distribution is compared with the observed climatology. In two experiments, a hypothetical Antarctic ozone hole is induced on October 1 and on November 1; the tracer model is integrated for 1 year with the standard linearized chemistry. The initial depletion, 90 percent of the O sub 3 poleward of 70 S between 25 and 180 mbar, amounts to about 5 percent of the total O sub 3 in the Southerm Hemisphere. As the vortex breaks down and the hole is dispersed, significant depletions to column ozone, of order 10 D.U., occur as far north as 36 S during Austral summer. One year later, about 25 percent of the original depletion remains, mostly below 100 mbar and poleward of 30 S. Details of the calculations are shown, along with a budget analysis showing the fraction of the hole filled in by photochemistry versus that transported into the troposhere

Interpolation inequalities and spectral estimates for magnetic operators

We prove magnetic interpolation inequalities and Keller-Lieb-Thir-ring estimates for the principal eigenvalue of magnetic Schr{\"o}dinger operators. We establish explicit upper and lower bounds for the best constants and show by numerical methods that our theoretical estimates are accurate

On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities

In this paper we prove some new symmetry results for the extremals of the Caffarelli-Kohn-Nirenberg inequalities, in any dimension larger or equal than two

Changing stroke mortality trends in middle-aged people: an age-period-cohort analysis of routine mortality data in persons aged 40 to 69 in England

Background: In the UK, overall stroke mortality has declined. A similar trend has been seen in coronary heart disease, although recent reports suggest this decline might be levelling off in middle-aged adults. Aim: To investigate recent trends in stroke mortality among those aged 40–69 years in England. Methods: The authors used routine annual aggregated stroke death and population data for England for the years 1979–2005 to investigate time trends in gender-specific mortalities for adults aged 40 to 69 years. The authors applied log-linear modelling to isolate effects attributable to age, linear ‘drift’ over time, time period and birth cohort. Results; Between 1979 and 2005, age-standardised stroke mortality aged 40 to 69 years dropped from 93 to 30 per 100 000 in men and from 62 to 18 per 100 000 in women. Mortality was higher in older age groups, but the difference between the older and younger age groups appears to have decreased over time for both sexes. Modelling of the data suggests an average annual reduction in stroke deaths of 4.0% in men and 4.3% in women, although this decrease has been particularly marked in the last few years. However, we also observed a relative rate increase in mortality among those born since the mid-1940s compared with earlier cohorts; this appears to have been sustained in men, which explains the levelling off in the rate of mortality decline observed in recent years in the younger middle-aged. Conclusions: If observed trends in middle-aged adults continue, overall stroke mortalities may start to increase again