A Maximum Principle at Infinity for Surfaces with Constant Mean Curvature in Euclidean Space

Maximum principles at infinity generalize Hopf's maximum principle for hypersurfaces with constant mean curvature in R ⁿ . We establish such a maximum principle for parabolic surfaces in R³ with nonzero constant mean curvature and bounded Gaussian curvature.     

一类完全非线性椭圆型方程粘性解的比较原理

讨论一类带有非局部积分项的完全非线性椭圆型方程半连续粘性解的比较原理，这类方程源自带跳跃的扩散过程，在随机控制，金融数学中有广泛而重要的应用。     

Regularity of Viscosity Solutions of the Biased Infinity Laplacian Equation

In this paper, we are interested in the regularity estimates of the nonnegative viscosity super solution of the $β$−biased infinity Laplacian equation $$∆^β_∞u = 0,$$ where $β ∈ \mathbb{R}$ is a fixed constant and $∆^β_∞u := ∆^N_∞u + β|Du|,$ which arises from the random game named biased tug-of-war. By studying directly the $β$−biased infinity Laplacian equation, we construct the appropriate exponential cones as barrier functions to establish a key estimate. Based on this estimate, we obtain the Harnack inequality, Hopf boundary point lemma, Lipschitz estimate and the Liouville property etc.     

A-调和方程很弱解的比较原理

应用McShane扩张定理证明了A-调和方程很弱解的比较原理.这个结果可以看作经典弱解比较原理的一个扩展.     

Limits at Infinity of Superharmonic Functions and Solutions of Semilinear Elliptic Equations of Matukuma Type

In an unbounded domain Ω in ℝ ⁿ (n ≥ 2) with a compact boundary or Ω = ℝ ⁿ , we investigate the existence of limits at infinity of positive superharmonic functions u on Ω satisfying a nonlinear inequality like as

Viscosity Solutions for the Two-Phase Stefan Problem

We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates possible existence of a mushy region generated by the initial data. We show that a comparison principle holds between viscosity solutions, and investigate the coincidence of the viscosity solutions and the weak solutions defined via integration by parts. In particular, in the absence of initial mushy region, viscosity solution is the unique weak solution with the same boundary data.     

Integral Representations,Differentiability Properties and Limits at Infinity for Beppo Levi Functions

The first aim in the present paper is to give an integral representation for Beppo Levi functions on R ⁿ. Our integral representation is an extension of Sobolev's integral representation given for infinitely differentiable functions with compact support. As applications, continuity and differentiability properties of Beppo Levi functions are studied.Our second aim in this paper is to study the existence of limits at infinity for Beppo Levi functions. We also consider the existence of fine-type limits at infinity with respect to Bessel capacities, which yields the radial limit result at infinity.     

Fast Summation of Power Series with Coefficients Analytic at Infinity

We propose a fast summation algorithm for slowly convergent power series of the form _{j=j 0} z ^j _j j _i=1 ^s (j+ _i )^– _i , where R, _i 0 and _i C, 1is, are known parameters, and _j =(j), being a given real or complex function, analytic at infinity. Such series embody many cases treated by specific methods in the recent literature on acceleration. Our approach rests on explicit asymptotic summation, started from the efficient numerical computation of the Laurent coefficients of . The effectiveness of the resulting method, termed ASM (Asymptotic Summation Method), is shown by several numerical tests.     

关于动态规划方程受约束粘性解的比较定理

证明了与随机控制问题有关的动态规划方程粘性解的比较定理 .该定理对研究一类随机金融控制问题起着重要的作用     

高阶脉冲时滞微分方程解的振动准则(英文)

本文研究了一类n阶线性脉冲时滞微分方程解的振动性。通过比较原理,得到了其振动的充分条件,所得到的结果推广了一些已有的结果。     

The Ends of Manifolds with Bounded Geometry, Linear Growth and Finite Filling Area

We prove that simply connected open manifolds of bounded geometry, linear growth and sublinear filling growth (e.g. finite filling area) are simply connected at infinity.     

A Property of Solutions for Elliptic Equation with Nonstandard Growth Conditions in Unbounded Domain

§０．　ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔΩｂｅａｎｕｎｂｏｕｎｄｅｄｄｏｍａｉｎｉｎｔｈｅｎ－ｄｉｍｅｎｓｉｏｎａｌＥｕｃｌｉｄｅａｎｓｐａｃｅＥｎ（ｎ３）．Ｃｏｎｓｉｄｅｒ－ｉｎｇｔｈｅｆｏｌｌｏｗｉｎｇｅｌｌｉｐｔｉｃｅｑｕａｔｉｏｎｗｉｔｈｎｏｎｓｔａｎｄａｒｄｇｒｏｗｔｈｃｏｎｄｉｔｉｏｎｓ：－ｄｉｖＡ（ｘ,ｕ,ｕ）＝Ｂ（ｘ,ｕ,ｕ）（１）ｗｈｅｒｅＡ（ｘ,ｕ,ξ）＝｛Ａｉ（ｘ,ｕ,ξ）,ｉ＝１,２,…,ｎ｝ａｎｄＢ（ｘ,ｕ,ξ）ａｒｅｄｅｆｉｎｅｄｏｎＥｎ×Ｅ１×Ｅｎ,ｃｏｎ－…     

非齐次A-调和方程弱解的局部极值原理

研究非齐次A-调和方程divA(x,▽u(x))=B(x,▽u(x)),其满足〈A(x,h),h〉≥α|h|~p,|A(x,h)|≤β|h|~(p-1)+g(x),|B(x,h)|≤γ|h|~(p-1).应用Moser迭代法证明其弱解满足局部极值原理,并进一步得出弱解的内部估计和全局估计.     

Continuous Dependence Estimates for Viscosity Solutions of Fully Nonlinear Degenerate Parabolic Equations

Using the maximum principle for semicontinuous functions (Differential Integral Equations3 (1990), 1001-1014; Bull. Amer. Math. Soc. (N.S)27 (1992), 1-67), we establish a general “continuous dependence on the non- linearities” estimate for viscosity solutions of fully nonlinear degenerate parabolic equations with time- and space-dependent nonlinearities. Our result generalizes a result by Souganidis (J. Differential Equations56 (1985), 345-390) for first- order Hamilton-Jacobi equations and a recent result by Cockburn et al. (J. Differential Equations170 (2001), 180-187) for a class of degenerate parabolic second-order equations. We apply this result to a rather general class of equations and obtain: (i) Explicit continuous dependence estimates. (ii) L^∞ and Hölder regularity estimates. (iii) A rate of convergence for the vanishing viscosity method. Finally, we illustrate results (i)-(iii) on the Hamilton-Jacobi- Bellman partial differential equation associated with optimal control of a degenerate diffusion process over a finite horizon. For this equation such results are usually derived via probabilistic arguments, which we avoid entirely here.     

Dominant and recessive solutions at infinity and genera of conjoined bases for discrete symplectic systems

In this paper we introduce the theory of dominant solutions at infinity for nonoscillatory discrete symplectic systems without any controllability assumption. Such solutions represent an opposite concept to recessive solutions at infinity, which were recently developed for such systems by the authors. Our main results include: (i) the existence of dominant solutions at infinity for all ranks in a given range depending on the order of abnormality of the system, (ii) construction of dominant solutions at infinity with eventually the same image, (iii) classification of dominant and recessive solutions at infinity with eventually the same image, (iv) limit characterization of recessive solutions at infinity in terms of dominant solutions at infinity and vice versa, and (v) Reid’s construction of the minimal recessive solution at infinity. These results are based on a new theory of genera of conjoined bases for symplectic systems developed for this purpose in this paper.     

Global Solution for Quasilinear Wave Equations with Viscosity and Nonlinear Perturbation

1 IntroductionIn this paper we study globaJ solutions to the initiaLboundary value prob1em for quasilinearwave equations with viscosity and a nonlinear perterbation of the fOrmwhere fl is a bounded domain in RN with smooth boundarY afl, and the norilineax termsa(vp), g(u) axe like a(vp) = (l + )vlP)--'/P(P > 1), g(u) =--ju1"u.This problem describes the motion of fixed membrane with strong viscosity. The globalexistence and stability Of smooth solutions fOr one space dimensional case N = 1…     

比较原理和带马尔可夫调制的随机泛函微分方程

本文考虑带马尔可夫调制的随机泛函微分方程解的不稳定性,通过建立的新的比较原理,得到一些不稳定的判据.     

具有临界增长的p-双调和方程非平凡解的存在性

本文在有界区域Ω■RN中讨论p-双调和方程△(a(x)|△u|p-2△u)=f(x,u)的Dirichlet 零边值问题,给出了在一般的临界增长条件下非平凡W02,p(Ω)解的存在性．     

One-Dimensional Riemann Problem for Equations of Constant Pressure Fluid Dynamics with Measure Solutions by the Viscosity Method

The Riemann problem for the equations of constant pressure fluid dynamics was considered. Solutions of this problem were constructed by employing the viscosity vanishing approach. For some initial data, solutions can be viewed as bounded functions in L(R × R+) plus bounded linear functionals on C0(R× R+) with nonclassical waves as their supports. Vacuum regions appear so that uniqueness of Riemann solutions fails for some initial data.