784 research outputs found
Sharp decay estimates for critical Dirac equations
No full textWe prove sharp pointwise decay estimates for critical Dirac equations on R^n with n ≥ 2. They appear for instance in the study of critical Dirac equations on compact spin manifolds, describing blow-up profiles, and as effective equations in honeycomb structures. For the latter case, we find excited states with an explicit asymptotic behavior. Moreover, we provide some classification results both for ground states and for excited states
Lieb–Thirring Inequality for the 2D Pauli Operator
No full textBy the Aharonov–Casher theorem, the Pauli operator P has no zero eigenvalue when the normalized magnetic flux α satisfies |α|0. Besides the usual semiclassical integral, the right side of our inequality involves an integral where the zero energy resonance state appears explicitly. Our inequality improves earlier works that were restricted to moments of order γ≥1
The generalized Wehrl entropy bound in quantitative form
No full textLieb and Carlen have shown that mixed states with minimal Wehrl entropy are coherent states. We prove that mixed states with almost minimal Wehrl entropy are almost coherent states. This is proved in a quantitative sense where both the norm and the exponent are optimal and the constant is explicit. We prove a similar bound for generalized Wehrl entropies. As an application, a sharp quantitative form of the log-Sobolev inequality for functions in the Fock space is provided
Blow-up of solutions of critical elliptic equations in three dimensions
No full textWe describe the asymptotic behavior of positive solutions u 6 of the equation - A u + au = 3u 5 - 6 in Q c R 3 with a homogeneous Dirichlet boundary condition. The function a is assumed to be critical in the sense of Hebey and Vaugon, and the functions u 6 are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of Brezis and Peletier (1989). Similar results are also obtained for solutions of the equation - A u + ( a + E V ) u = 3u 5 in Q
Weak perturbations of the p-Laplacian
No full textWe consider the p-Laplacian in R^d perturbed by a weakly coupled potential. We calculate the asymptotic expansions of the lowest eigenvalue of such an operator in the weak coupling limit separately for p>d and p=d and discuss the connection with Sobolev interpolation inequalities
The Walverton Senate of Delta Theta Phi, including Frank Healy, Roy Lockenour, Ray L. Smith, Rupert Park
No full textBlack and WhitePeople: Healy, Frank; Lockenour, Roy M.; Smith, Ray L.; Park, Ruper
The BCS Critical Temperature in a Weak External Electric Field via a Linear Two-Body Operator
No full textWe study the critical temperature of a superconductive material in a weak external electric potential via a linear approximation of the BCS functional. We reproduce a similar result as in Frank et al. (Commun Math Phys 342(1):189–216, 2016, [5]) using the strategy introduced in Frank et al. (The BCS critical temperature in a weak homogeneous magnetic field, [2]), where we considered the case of an external constant magnetic field
Energy asymptotics in the Brezis–Nirenberg problem: The higher-dimensional case
No full textInternational audienceFor dimensions N ě 4, we consider the Brézis-Nirenberg variational problem of finding Sp V q :" inf 0ıuPH 1 0 pΩq ş Ω |∇u| 2 dx` ş Ω V |u| 2 dx ş Ω |u| q dx¯2 {q , where q " 2N N´2 is the critical Sobolev exponent, Ω Ă R N is a bounded open set and V : Ω Ñ R is a continuous function. We compute the asymptotics of Sp0q´Sp V q to leading order as Ñ 0`. We give a precise description of the blow-up profile of (almost) minimizing sequences and, in particular, we characterize the concentration points as being extrema of a quotient involving the Robin function. This complements the results from our recent paper in the case N " 3
Energy asymptotics in the three-dimensional Brezis–Nirenberg problem
No full textFor a bounded open set we consider the minimization problem
involving the critical Sobolev exponent. The function is assumed to be critical in the sense of Hebey and Vaugon. Under certain assumptions on and we compute the asymptotics of as , where is the Sobolev constant. (Almost) minimizers concentrate at a point in the zero set of the Robin function corresponding to and we determine the location of the concentration point within that set. We also show that our assumptions are almost necessary to have
Discontinuities in the Distribution of Great Wealth: Sectoral Forces Old and New
No full textNational surveys of household economics and well-being in the United States usually focus on income. In those income surveys with supplemental wealth modules, the very rich are underrepresented if not unrepresented. Typically, wealth data are truncated such that they do not afford a view of the extreme top of the distribution. Therefore, we attempt to supplement our knowledge about elite wealth holdings by compiling data on the richest individuals and families in the United States. To do so, we draw from the rosters of the "Forbes Four Hundred," which have been published annually by Forbes magazine since 1982. Along with information from other business press reports and standard biographical sources, rosters of the very rich enable research on inequality at the extreme of the wealth distribution during a period of dramatic change in the composition and concentration of wealth. In this study, we focus analytically on economic sectors because we are interested less in the maldistribution of wealth by demographic groups than in inequality between different economic sectors. We will first specify our analytical approach, then examine issues in the use of business press rosters of the very rich as a data source, and follow with a discussion of the dimensions and categories of our sector typology. After presenting our results, we will address how sectoral forces old and new affect economic opportunity and great wealth outcomes.
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