A monotonicity theorem and its application to stationary solutions of the phase field model

In this paper the authors prove a monotonicity theorem for solutionsof a boundary value problem and use it to investigate the behaviourof stationary solutions of the one-dimensional phase field modelas , which measures the range of intermolecular forces, goesto zero.     

A monotonicity theorem and its applications to weighted elliptic equations

We study the equation w_(tt) + ?_(S~(N-1))w-μw_t-δw + h(t, ω)w~p= 0,(t, ω) ∈ R × S~(N-1), and under some conditions we prove a monotonicity theorem for its positive solutions. Applying this monotonicity theorem,we obtain a Liouville-type theorem for some nonlinear elliptic weighted singular equations. Moreover, we obtain the necessary and sufficient condition for-div(|x|~θ▽u) = |x|~lu~p, x ∈ R~N\{0} having positive solutions which are bounded near 0, which is also a positive answer to Souplet's conjecture(see Phan and Souplet(2012)) on the weighted Lane-Emden equation-?u = |x|~au~p, x ∈ R~N.     

A uniqueness theorem for weak solutions of the stationary semiconductor equations

In this paper we prove a uniqueness theorem for weak solutions of a mixed boundary-value problem for the stationary semiconductor equations (van Roosbroeck's system) under the assumption that the deviation of the carrier potentials from an equilibrium solution is sufficiently small.     

拓扑定理及其在超线性脉冲方程中的应用

本文研究一类含脉冲的非保守超线性方程的周期振动问题.通过详细刻画跳跃映射的角度函数,并应用相平面分析方法,本文证明方程的Poincaré映射有无穷多个不动点,而这些不动点对应了方程的无穷多个2π周期解.由于本文考虑的Poincaré映射不是同胚,所以,本文重新构造并证明了一个拓扑定理用于替代经典的Poincaré-Birkhoff扭转不动点定理.     

Stationary solutions to phase field crystal equations

This paper is devoted to the analytical and numerical study of stationary solutions of the one‐dimensional Phase Field Crystal Equation. This new model recently introduced by K. Elder and M. Grant describes phase transformations at the atomic level on large time scales. By using bifurcation methods, we investigate the quantitative and qualitative properties of these solutions: multiplicity, stability, and periodicity. Quite unusual bifurcation diagrams are obtained by numerical simulations. Copyright © 2010 John Wiley & Sons, Ltd.     

非守恒相场模型解的blow-up

讨论了一类非守恒相场模型解的性态,证明当ａ２ｐ － １ ＜ ０ 及初值充分大时解在有限时刻 ｂｌｏｗ ｕｐ．     

A density theorem and its application

We compute the density of primes represented by a special quadratic form in a fixed square residue class. Using this result and a new method introduced by Thaine we prove the fact that for a prime p 3congruent to 3 modulo 4, the component e(p+1)/2of the p-Sylow subgroup of the ideal class group of Q(ζp) is trivial.     

一个因式相通定理及其应用

证明了一个因式相通定理,它是奇点理论中等价性定理的一个推广.这个定理表明单纯高阶项的开折对普适开折因式相通中的参数映射没有贡献.据此本文提出了一种因式相通中参数映射的简洁有效的近似计算方法     

Symmetry and monotonicity of least energy solutions

We give a simple proof of the fact that for a large class of quasilinear elliptic equations and systems the solutions that minimize the corresponding energy in the set of all solutions are radially symmetric. We require just continuous nonlinearities and no cooperative conditions for systems. Thus, in particular, our results cannot be obtained by using the moving planes method. In the case of scalar equations, we also prove that any least energy solution has a constant sign and is monotone with respect to the radial variable. Our proofs rely on results in Brothers and Ziemer (J Reine Angew Math 384:153–179, 1988) and Mariş (Arch Ration Mech Anal, 192:311–330, 2009) and answer questions from Brézis and Lieb (Comm Math Phys 96:97–113, 1984) and Lions (Ann Inst H Poincaré Anal Non Linéaire 1:223–283, 1984).     

A general impossibility theorem and its application to individual rights

In this paper, we generalize Green and Laffont’s (1979) impossibility theorem to the following form: in quasi-linear environments, when the set of each agent’s types is sufficiently rich, we cannot find mechanisms that allow bounded deviations from the decisive efficiency, incentive compatibility and budget-balance at the same time. Hence, it is impossible to find an incentive compatible mechanism with minimum social welfare losses. Furthermore, we discuss the compatibility problems between incentive and individual rights in a quasi-linear environment (see Sen, 1970a,b; Deb et al., 1997). Specifically, some new impossibility results are established.     

Local monotonicity properties of two-variable Gini means and the comparison theorem revisited

In this paper we examine local monotonicity properties of the so-called Gini means, defined by

     

Multiplicity of stationary solutions to the Euler-Poisson equations

Consider the system of Euler-Poisson as a model for the time evolution of gaseous stars through the self-induced gravitational force. We study the existence, uniqueness and multiplicity of stationary solutions for some velocity fields and entropy function that solve the conservation of mass and energy a priori. These results generalize the previous works on the irrotational or the rotational gaseous stars around an axis, and then they hold in more general physical settings. Under the assumption of radial symmetry, the monotonicity properties of the radius of the gas with respect to either the strength of the velocity field or the center density are also given which yield the uniqueness under some circumstances.     

Existence of stationary solutions to the Vlasov-Poisson-Boltzmann system

In this paper, we study the existence of stationary solutions to the Vlasov-Poisson-Boltzmann system when the background density function tends to a positive constant with a very mild decay rate as |x|→∞. In fact, the stationary Vlasov-Poisson-Boltzmann system can be written into an elliptic equation with exponential nonlinearity. Under the assumption on the decay rate being (ln(e+|x|))^−α for some α>0, it is shown that this elliptic equation has a unique solution. This result generalizes the previous work [R. Glassey, J. Schaeffer, Y. Zheng, Steady states of the Vlasov-Poisson-Fokker-Planck system, J. Math. Anal. Appl. 202 (1996) 1058-1075] where the decay rate is assumed.     

A theorem on parastatistics and its application

Proceeding from the assumption that nature chooses a riskless strategy, we consequently obtain a distribution over parastatistics. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 3, pp. 511–512, March, 2007.     

Convergence rate of solutions toward stationary solutions to a two-phase model with magnetic field in a half line

In this paper, we study an asymptotic behavior of a solution to the outflow problem for a two-phase model with magnetic field. Our idea mainly comes from [1] and [2] which investigate the asymptotic stability and convergence rates of stationary solutions to the outflow problem for an isentropic Navier–Stokes equation. For the two-phase model with magnetic field, we also obtain the asymptotic stability and convergence rates of global solutions towards corresponding stationary solutions if the initial perturbation belongs to the weighted Sobolev space. The proof is based on the weighted energy method.