On Singular Sets of Local Solutions to p-Laplace Equations

The author proves that the right-hand term of a p-Laplace equation is zero on the singular set of a local solution to the equation. Such a result is used to study the existence of an optimal control problem.     

一维奇异p-Laplace方程的正解

本文获得了一维奇异p－Laplace方程的Dirichlet边值问题（|y′|p-2y′）′＋g（t，y）＝0，y（0）＝y（1）＝0，P＞1的两个正解存在定理．这一结论是通过使用Leray－Schauder非线性抉择及上下解方法建立的．     

Some Solutions for a Class of Singular Equations

In this paper we obtain all solutions which depend only on r for a class of partial differential equations of higher order with singular coefficients.     

On the Uniqueness of Solution for a Kind of Singular Integral Equations

In this article, from the discussion of some properties of entire functions, the uniqueness of solution for a kind of singular integral equations with multiple complex translations on the real axis is proved.     

关于奇异非线性多调和方程的正整体解

该文主要研究形如Δ((Δ\+nu)\+\{p-1*\}) = f(|x|, u, |u|)u\+\{-β\},\ x∈R\+2的奇异非线性多调和方程在R\+2上的正整体解，此处p>1，β≥0是常数，n是自然数，f: [AKR-]\-+×R\-+×[AKR-]\-+→R\-+是 一个连续函数,ξ\+\{α*\}:=|ξ|\+\{α-1\}ξ,ξ∈R，α>0 . 证明了这种解 u必无界且其渐进阶（当n→∞时u作为无穷大量的阶）不低于|x|\+\{2n\}log|x| ，给 出了该方程具有无穷多个其渐进阶刚好为 |x|\+\{2n\}log|x| 的正整体解的充分与充分必要条件. 这些结论可以推广到更一般的方程中去.      

Some Optimization Problems for <Emphasis Type="Italic">p</Emphasis>-Laplacian Type Equations

In this paper we study some optimization problems for nonlinear elastic membranes. More precisely, we consider the problem of optimizing the cost functional over some admissible class of loads f where u is the (unique) solution to the problem −Δ _p u+|u| ^p−2 u=0 in Ω with |∇ u| ^p−2 u _ν =f on ∂Ω. Supported by Universidad de Buenos Aires under grant X078, by ANPCyT PICT No. 2006-290 and CONICET (Argentina) PIP 5478/1438. J. Fernández Bonder is a member of CONICET. Leandro M. Del Pezzo is a fellow of CONICET.     

Second-Order Optimality Conditions for Singular Pontryagin Local Minimizers

In this article, we use the method of finite-dimensional approximations to derive second-order necessary optimality conditions for singular Pontryagin local minimizers for an unconstrained nonlinear Mayer problem.     

脉冲集值微分方程解的存在性

利用Leray-Schauder不动点定理,证明一类一阶脉冲集值微分方程解的存在性.     

On the Cauchy Problem of Evolution P-Laplacian Equations with Strongly Non-linear Sources When 1 < p < 2

We study the solvability of the Cauchy problem (1.1)–(1.2) for the largest possible class of initial values, for which (1.1)–(1.2) has a local solution. Moreover, we also study the critical case related to the initial value u ₀, for 1 < p < ∞. Project supported by the National Natural Science Foundation of China (19971070)     

On the p-Ranks and Characteristic Polynomials of Cyclic Difference Sets

In this paper, the p-ranks and characteristic polynomials of cyclic difference sets are derived by expanding the trace expressions of their characteristic sequences. Using this method, it is shown that the 3-ranks and characteristic polynomials of the Helleseth–Kumar–Martinsen (HKM) difference set and the Lin difference set can be easily obtained. Also, the p-rank of a Singer difference set is reviewed and the characteristic polynomial is calculated using our approach.     

一类带奇异系数椭圆方程解的存在性

本文运用变分方法及Hardy不等式讨论了一类带奇异系数的临界椭圆方程，证明了在一定条件下方程解的存在性。     

一类奇异半线性热方程初值问题解 的唯一性结果

设ｕ（ｔ,ｘ）,ｕ（ｔ,ｘ）为初值问题在带形域ＳＴ＝（０,Ｔ）×Ｒｎ内的两个非负经曲解,ｆ（ｘ）连续有界非负的实函数,则有如下的结果：（１）若ｆ（ｘ）不恒为零,则在ＳＴ中ｕ（ｔ,ｘ）;（２）若γ＞１,则在ＳＴ中ｕ（ｔ,ｘ）ｕ（ｔ,ｘ）;（３）若０＞γ＞１,ｆ（ｘ）０,则问题（１．１）,（１．２）的解不唯一且它的所有非平凡解的集合为ｕ（ｔ,ｓ）＝这里ｓ≥０是参数,其中记号（γ）＋＝ｍａｘ｛γ,０｝．     

Capacity estimate for the blow-up set of parabolic equations

A capacity estimate for the blow-up set of parabolic equations is derived. It refines the Lebesgue measure estimate (Sakaguchi and Suzuki in Arch Rational Mech Anal 142:143–153, 1998), includes the result on the elliptic case (T. Sato, T. Suzuki, F. Takahashi, in p-capacity of the singular set of p-harmonic function vanishes, preprint), and provides information on the profile of any post-blow-up solution.      

On Integer Solutions to Linear Equations

A magic square is an n × n matrix with non-negative integer entries, such that the sum of the entries in each row and column is the same. We study the enumeration and P-recursivity of these in the case in which the sum along each row and column is fixed, with the size n of the matrix as the variable. A method is developed that nicely proves some known results about the case when the row and column sum is 2, and we prove new results for the case when the sum is 3. Received December 23, 2005     

具超临界指数p-Laplace方程解的存在性

证明了一类具超临界指数p-Laplace方程在全空间和有界域上正解的存在性,给出了解的一些估计.     

关于《一类奇异边值问题的正解》的注记

文[4]通过构造反例断言文[1]中定理的必要性证明有误,本文首先指出文[4] 的这个断言不正确,然后对文[4]中定理2.1作了本质性的改进.     

A Remark on the Existence of Solutions to Nonlinear Equations with Degenerate Mappings

The paper studies nonlinear mappings using methods of p-regularity theory and some concepts and technics of set-valued analysis. The main result addresses to the problem of existence of solutions to nonlinear equations in the degenerate case where a linear part may be singular at the considered initial point.      

Global Estimates for Non-symmetric Green Type Functions with Applications to the p-Laplace Equation

This paper presents global estimates for non-symmetric Green type functions, which are applicable to singular functions for the p-Laplace equation. This work was partially supported by Grant-in-Aid for Young Scientists (B) (No. 19740062), Japan Society for the Promotion of Science.     

Asymptotic Behavior of Solutions to a Class of Systems of Delay Differential Equations

The authors investigate the asymptotic behavior of solutions to a class of systems of delay differential equations. It is shown that every bounded solution of such a class of systems tends to a constant vector as t→∞. Our results improve and extend some corresponding ones already known.