Symmetry results for overdetermined boundary value problems of nonlinear elliptic equations

We prove the radial symmetry of the solutions of second-order nonlinear elliptic equations for overdetermined Dirichlet and Neumann boundary value problems. In addition, a global uniqueness theorem of Holmgren type is given for nonlinear elliptic equations.     

Null boundary controllability for semilinear heat equations

We consider null boundary controllability for one-dimensional semilinear heat equations. We obtain null boundary controllability results for semilinear equations when the initial data is bounded continuous and sufficiently small. In this work we also prove a version of the nonlinear Cauchy-Kowalevski theorem.W. Littman was partially supported by NSF Grant DMS 90-02919. The results of this paper were presented by Yung-Jen Lin Guo at the P.D.E. seminar at the University of Minnesota on January 27, 1993 and by W. Littman at the First International Conference on Dynamics Systems and Applications held in Atlanta in May 1993.     

The periodic Cauchy problem of the modified Hunter-Saxton equation

We prove that the periodic initial value problem for the modified Hunter-Saxton equation is locally well-posed for initial data in the space of continuously differentiable functions on the circle and in Sobolev spaces when s > 3/2. We also study the analytic regularity (both in space and time variables) of this problem and prove a Cauchy-Kowalevski type theorem. Our approach is to rewrite the equation and derive the estimates which permit application of o.d.e. techniques in Banach spaces. For the analytic regularity we use a contraction argument on an appropriate scale of Banach spaces to obtain analyticity in both time and space variables.     

Computing the region of convergence for power series in many real variables: A ratio-like test

We give an elementary proof that the region of convergence for a power series in many real variables is a star-convex domain but not, in general, a convex domain. In doing so, we deduce a natural higher-dimensional analog of the so-called ratio test from univariate power series. From the constructive proof of this result, we arrive at a method to approximate the region of convergence up to a desired accuracy. While most results in the literature are for rather specialized classes of multivariate power series, the method devised here is general. As far as applications are concerned, note that while theorems such as the Cauchy-Kowalevski theorem (and its generalizations to many variables) grant the existence of a region of convergence for a multivariate Taylor series to certain PDEs under appropriate restrictions, they do not give the actual region of convergence. The determination of the maximal region of convergence for such a series solution to a PDE is one application of our result.     

Cauchy-Kowalevski Extensions and Monogenic Plane Waves in Clifford Analysis

This paper deals with axially and biaxial monogenic functions that are derived using two fundamental methods of Clifford analysis, namely the Cauchy-Kowalevski extension and monogenic plane waves.     

A Noncharacteristic Cauchy Problem for Linear Parabolic Equations I: Solvability

Noncharacteristic Cauchy problems for parabolic equations arc frequently encountered in many areas of heat transfer. These problems are well known to be severely ill-posed. In this paper a solvability criterion for a class of such problems is established. It is proved that a weak solution of a noncharacteristic Cauchy problem for linear parabolic equations in divergence form with coefficients in a Holmgren class 2 in time exists if and only if the Cauchy data arc functions of a Holmgren class 2! A function g(t) defined on (α, β) is said to be of a Holmgren class 2, if g ?C^∞ (α, β) and for all nonnegative integers n there exist positive constants c and s such that |g⁽ⁿ⁾| < csⁿ(2n)!.     

Fixed points of approximable maps

We present a simple proof of the Leray-Schauder type theorem for approximable multimaps given recently by Ben-El-Mechaiekh and Idzik. We apply this theorem to obtain a Schaefer type theorem, the Birkhoff-Kellogg type theorems, a Penot type theorem for non-self-maps, and quasi-variational inequalities, all related to compact closed approximable maps.

     

一个新的Ky Fan匹配定理及其在L-凸度量空间中的应用

建立了转移开覆盖的一个新的Ky Fan型匹配定理．作为应用，获得了Fan-Browder重合定理，不动点定理、极大元定理，相交定理和截口定理．     

紧开覆盖的一个新的Ky Fan型匹配定理及其应用

建立了紧开覆盖的一个新的Ky Fan型匹配定理.作为应用,研究了Fan-Browder型重合定理、Ky Fan型最佳逼近定理、极大元定理及超凸空间中的定性对策和抽象经济的平衡存在定理.     

Scalar derivatives and scalar asymptotic derivatives. An Altman type fixed point theorem on convex cones and some applications

An Altman type fixed point theorem, on a convex cone in a Banach space, will be presented. In this theorem we replace the classical Altman's condition by an asymptotic condition. Also, we will establish some relations between this Altman type fixed point theorem and the scalar derivatives. Some applications will be given.     

Korovkin type theorem for non‐tensor Balázs type Bleimann,Butzer and Hahn operators

In this paper, we prove a certain Korovkin type approximation theorem by introducing new test functions. We introduce the non‐tensor Balázs type Bleimann, Butzer and Hahn operators and give the approximation property by using this new Korovkin theorem. Furthermore, we obtain the rate of convergence of these operators by means of modulus of continuity. Finally, we state the multivariate version of the abovementioned Korovkin type theorem. Copyright © 2014 John Wiley & Sons, Ltd.     

THE MORSE CRITICAL GROUPS OF THE MINMAX THEOREM

The author gives a new min-max theorem and studies the local homology propertiesof critical points that is got by min-max theorem.In particular,this result includes manymin-max theorem of link type,which can be applicable to nonlinear analysis.     

A refinement of Sharkovskii's theorem on orbit types characterized by two parameters

The so-called type problem or forcing problem is considered as a way to generalize Sharkovskii's theorem. In this paper, by focusing on certain types of orbits, we obtain a solution of the type problem, which gives a refinement of Sharkovskii's theorem on orbit types characterized by two parameters.     

MINIMAX PRINCIPLES FOR FUNCTIONS WITH TRANSFER LOWER SEMI-CONTINUITY AND APPLICATIONS

本文引入了转移下半连续的概念并在转移下半连续的条件下推广了ＫｙＦａｎ极大极小原理。应用一这结果，证明了截口定理、不动点定理和函数不等式组解的存在性定理。     

Cauchy-Kowalevski extensions and monogenic plane waves using spherical monogenics

Clifford analysis may be regarded as a direct and elegant generalization to higher dimensions of the theory of holomorphic functions in the complex plane, centred around the notion of monogenic function, i.e. a null solution of the Dirac operator. This paper dealswith axial and biaxialmonogenic functions containing sphericalmonogenics. They are constructed bymeans of two fundamentalmethods of Clifford analysis, namely the Cauchy-Kowalevski extension and monogenic plane waves.     

齐次群上二阶半线性偏微分方程的一类Picone型恒等式和Sturmian比较定理

本文给出了齐次群上的一类广义Picone型恒等式,由此证明了以下半线性方程组(其中 表示齐次群上的广义梯度)的Sturmian比较定理及一类振荡定理,并用于Heisenberg群上一类半线性方程.然后利用这里的广义Picone型恒等式证明了Heisenberg群上一类更一般的Hardv型不等式     

A Radon–Nikodym type theorem for forms

The Lebesgue decomposition theorem and the Radon–Nikodym theorem are the cornerstones of the classical measure theory. These theorems were generalized in several settings and several ways. Hassi, Sebestyén, and de Snoo recently proved a Lebesgue type decomposition theorem for nonnegative sesquilinear forms defined on complex linear spaces. The main purpose of this paper is to formulate and prove also a Radon–Nikodym type result in this setting. As an application, we present a Lebesgue type decomposition theorem and solve a special case of the infimum problem for densely defined (not necessarily bounded) positive operators.     

Derivation fonctionnelle abstraite

In this paper, the classical theory, due to R. de POSSEL [11][12], of abstract derivation basis with VITALI properties, is systematically generalized with a functional formulation. That is to say that the derivation by means of suitable families of measurable sets is replaced by a derivation using measurable functions. A theorem which gives various caracterizations of that functional basis of weak VITALI type includes a theorem obtained by R. de POSSEL [13] about this kind of derivation. This result leads to the introduction of VITALI conditions of functional type in a derivation theorem of N. DUNFORD and J. T. SCHWARTZ [3].     

非紧L-凸度量空间中的GLsKKM定理及其对不动点问题的应用

建立了非紧完备L-凸度量空间中的GLsKKM定理.作为应用,获得了Ky Fan 截口定理,极大元定理和Fan-Browder型不动点定理.     

Bitsadze–Samarskii Boundary Condition for Elliptic-Parabolic Volume Potential

A spectral decomposition of the Green’s function of the Holmgren problem in a cylindrical domain is used to obtain Bitsadze–Samarskii boundary conditions for a regular elliptic-parabolic volume potential.