Surface integral and Gauss-Ostrogradskij theorem from the viewpoint of applications

Making use of a surface integral defined without use of the partition of unity, trace theorems and the Gauss-Ostrogradskij theorem are proved in the case of three-dimensional domains with a Lipschitz-continuous boundary for functions belonging to the Sobolev spaces H ^1,p () (1 p < ). The paper is a generalization of the previous author's paper which is devoted to the line integral.     

Lattice polygons and Green's theorem

Associated to an -dimensional integral convex polytope is a toric variety and divisor , such that the integral points of represent . We study the free resolution of the homogeneous coordinate ring as a module over . It turns out that a simple application of Green's theorem yields good bounds for the linear syzygies of a projective toric surface. In particular, for a planar polytope , satisfies Green's condition if contains at least lattice points.

     

An estimate for the three-dimensional discrete Green's function and applications

In this paper we first introduce definitions of the regularized Green's function and the discrete Green's function in three dimensions. With complicated arguments estimates for these two functions are then derived. Finally we give an application of the estimate for the discrete Green's function to the high accuracy analysis of the three-dimensional block finite element approximation.     

Lagrange multipliers for functions derivable along directions in a linear subspace

We prove a Lagrange multipliers theorem for a class of functions that are derivable along directions in a linear subspace of a Banach space where they are defined. Our result is available for topological linear vector spaces and is stronger than the classical one even for two-dimensional spaces, because we only require the differentiablity of functions at critical points. Applying these results we generalize the Lax-Milgram theorem. Some applications in variational inequalities and quasilinear elliptic equations are given.

     

The generalized decomposition theorem in Banach spaces and its applications

In this paper, we first give a simple proof of the decomposition theorem in Alber (Field Inst. Comm. 25 (2000) 77) and then present a new decomposition of arbitrary elements in reflexive strictly convex and smooth Banach spaces. As applications of the decomposition theorem, we give the representations of the metric projection operator for some kind of closed convex sets. Finally, we provide a sufficient condition under which the generalized projection operator coincides with the metric projection operator.     

A FURTHER GENERALIZATION OF THE KKM THEOREM WITH APPLICATIONS

ＡＦＵＲＴＨＥＲＧＥＮＥＲＡＬＩＺＡＴＩＯＮＯＦＴＨＥＫＫＭＴＨＥＯＲＥＭ　ＷＩＴＨＡＰＰＬＩＣＡＴＩＯＮＳＪＩＡＮＧＪＩＡＨＥ（江嘉禾）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍｏｔｉｃｓ，ＵｎｉｖｅｒｓｉｔｙｏｆＥｌｅｃｔｒｏｎｉｃＳｃｉｅｎｃｅｃｕ...     

Fountain theorem over cones and applications

In this paper, we establish fountain theorems over cones and apply it to the quasilinear elliptic problem{-pu = λ|u|q-2u + μ|u| γ-2 u, x ∈Ω,u = 0, x ∈Ω,(1)to show that problem (1) possesses infinitely many solutions, where 1 p N, 1 q p γ, ΩRN is a smooth bounded domain and λ, μ∈ R.     

The basic theorem of complementarity revisited

The basic theorm of (linear) complementarity was stated in a 1971 paper [6] by B.C. Eaves who credited C.E. Lemke for giving a constructive proof based on his almost complementary pivot algorithm. This theorem asserts that associated with an arbitrary linear complementarity problem, a certain augmented problem always possesses a solution. Many well-known existence results pertaining to the linear complementarity problem are consequences of this fundamental theorem.In this paper, we explore some further implications of the basic theorem of complementarity and derive new existence results for the linear complementarity problem. Based on these results, conditions for the existence of a solution to a linear complementarity problem with a fully-semimonotone matrix are examined. The class of the linear complementarity problems with aG-matrix is also investigated.The work of this author was based on research supported by the National Science Foundation under grant ECS-8717968.     

A uniqueness theorem for a free boundary problem

In this paper, we prove a uniqueness theorem for a free boundary problem which is given in the form of a variational inequality. This free boundary problem arises as the limit of an equation that serves as a basic model in population biology. Apart from the interest in the problem itself, the techniques used in this paper, which are based on the regularity theory of variational inequalities and of harmonic functions, are of independent interest, and may have other applications.

     

A Variational Problem for Steady Vortices in a Shear Flow Past an Obstacle

We prove existence of maximizers for a variational problem for a steady vortex anomaly of bounded extent in a shear flow, past an obstacle, in a planar exterior domain. Kinetic energy is maximized subject to the vorticity being a rearrangement of a prescribed function.     

On the rigidity theorem for elliptic genera

We give a detailed proof of the rigidity theorem for elliptic genera. Using the Lefschetz fixed point formula we carefully analyze the relation between the characteristic power series defining the elliptic genera and the equivariant elliptic genera. We show that equivariant elliptic genera converge to Jacobi functions which are holomorphic. This implies the rigidity of elliptic genera. Our approach can be easily modified to give a proof of the rigidity theorem for the elliptic genera of level .

     

A mean-value theorem and its applications

For a function f defined in an interval I, satisfying the conditions ensuring the existence and uniqueness of the Lagrange mean L[f], we prove that there exists a unique two variable mean M[f] such that

     

An application of Schauder's fixed point theorem with respect to higher order BVPs

We shall provide conditions on the function . The higher order boundary value problem

has at least one solution.

     

A drop theorem without vector topology

Daneš' drop theorem is extended to bornological vector spaces. An immediate application is to establish Ekeland-type variational principle and its equivalence, Caristi fixed point theorem, in bornological vector spaces. Meanwhile, since every locally convex space becomes a convex bornological vector space when equipped with the canonical von Neumann bornology, Qiu's generalization of Daneš' work to locally convex spaces is recovered.     

Resolvent kernels of Green's function kernels and other finite-rank modifications of Fredholm and Volterra kernels

Many important Fredholm integral equations have separable kernels which are finite-rank modifications of Volterra kernels. This class includes Green's functions for Sturm-Liouville and other two-point boundary-value problems for linear ordinary differential operators. It is shown how to construct the Fredholm determinant, resolvent kernel, and eigenfunctions of kernels of this class by solving related Volterra integral equations and finite, linear algebraic systems. Applications to boundary-value problems are discussed, and explicit formulas are given for a simple example. Analytic and numerical approximation procedures for more general problems are indicated.This research was sponsored by the United States Army under Contract No. DAA29-75-C-0024.     

Existence and Properties of Radial Solutions of a Sub-linear Elliptic Equation

A non lipschitzian nonlinear elliptic equation is reviewed and results of existence, uniqueness, positivity and classification are proved using direct methods derived from the equation.     

Stokes' theorem for domains with fractal boundary

In this paper, integrals of second kind over a rectifiable curve or a piecewise smooth surface are extended to continuous fractal curves and surfaces. Theorems for the existence of these integrals are proved. Green's, Gauss' and Stokes' theorems are developed for domains with fractal boundaries.     

对称向量拟均衡问题有效解的存在性

利用标量化方法建立对称向量拟均衡问题有效解的存在性定理.作为标量化方法的应用,利用这一方法得到向量变分不等式和拟向量变分不等式有效解的存在性定理.