A new technique of integral representations in ?n

A new technique of integral representations in ℂ ⁿ , which is different from the well-known Henkin technique, is given. By means of this new technique, a new integral formula for smooth functions and a new integral representation of solutions of the ∂-equations on strictly pseudoconvex domains in ℂ ⁿ are obtained. These new formulas are simpler than the classical ones, especially the solutions of the ∂-equations admit simple uniform estimates. Moreover, this new technique can be further applied to arbitrary bounded domains in ℂ ⁿ so that all corresponding formulas are simplified.     

A new Leray formula for smooth functions on bounded domains in Cn

By means of a new technique of integral representations in Cn given by the authors, we establish a new abstract formula with a vector function W for smooth functions on bounded domains in Cn, which is different from the well-known Leray formula, This new formula eliminates the term that contains the parameter λ from the classical Leray formula, and especially on some domains the uniform estimates for the -equation are very simple. From the new Leray formula, we can obtain correspondingly many new formulas for smooth functions on many domains in Cn, which are different from the classical ones, when we properly select the vector function W.     

Singular integral on bounded strictly pseudoconvex domain

Kytmanov and Myslivets gave a special Cauchy principal value of the singular integral on the bounded strictly pseudoconvex domain with smooth boundary. By means of this Cauchy integral principal value, the corresponding singular integral and a composition formula are obtained. This composition formula is quite different from usual ones in form. As an application, the corresponding singular integral equation and the system of singular integral equations are discussed as well.     

A new Leray formula for smooth functions on bounded domains in Cn

By means of a new technique of integral representations in C ⁿ given by the authors, we establish a new abstract formula with a vector function W for smooth functions on bounded domains in C ⁿ , which is different from the well-known Leray formula. This new formula eliminates the term that contains the parameter A from the classical Leray formula, and especially on some domains the uniform estimates for the are very simple. From the new Leray formula, we can obtain correspondingly many new formulas for smooth functions on many domains in C ⁿ , which are different from the classical ones, when we properly select the vector function W.     

A family of integral representations for the Brownian variables

The natural filtration of a real Brownian motion and its excursion filtration are sharing a fundamental property: the property of integral representation. As a consequence, every Brownian variable admits two distinct integral representations. We show here that there are other integral representations of the Brownian variables. They make use of a stochastic flow studied by Bass and Burdzy. Our arguments are inspired by Rogers and Walsh's results on stochastic integration with respect to the Brownian local times.     

Construction of rigid local systems and integral representations of their sections

We give a method for constructing all rigid local systems of semi‐simple type, which is different from the Katz–Dettweiler–Reiter algorithm. Our method follows from the construction of Fuchsian systems of differential equations with monodromy representations corresponding to such local systems, which give an explicit solution of the Riemann–Hilbert problem. Moreover, we show that every section of such local systems has an integral representation. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sobolev-type integral representations for functions defined on Carnot groups

We obtain some integral representations of the form f(x) = P(f) + K(?f) on the Carnot groups, where P(f) is a polynomial and K is an integral operator with a specific singularity. These representations are employed to prove the weak Poincaré inequality.     

锥中调和函数的积分表示

本文证明了锥内一类调和函数h, 若其正部h⁺ = max{h,0} 满足一种增长条件, 则h 能被其边界值的积分表示. 同时证明了其负部h^- = max{-h,0} 也能被类似的一种增长条件所控制. 所得结论推广了解析函数和调和函数在上半空间中关于积分表示的相关结果.     

Uniqueness of smooth radial balanced metrics on the disc

We show that the usual Poincaré metric is the only radial balanced metric on the disc with not too wild boundary behaviour. Additionally, we identify explicitly all radial metrics with such boundary behaviour which satisfy the balanced condition as far as germs at the boundary are concerned. Related results for the annulus and the punctured disc are also established.     

Hardy-Hilbert积分不等式的一个新推广及其应用

通过引入一个形如x1 x(x∈[0, ∞))的幂指函数建立了带权的Hardy-Hilbert积分不等式的新推广.并证明了系数(2)(sinπp)是最佳值.作为应用,给出了Hardy-Littlewood积分不等式的一个推广.     

A new stabilizing technique for boundary integral methods for water waves

Boundary integral methods to compute interfacial flows are very sensitive to numerical instabilities. A previous stability analysis by Beale, Hou and Lowengrub reveals that a very delicate balance among terms with singular integrals and derivatives must be preserved at the discrete level in order to maintain numerical stability. Such balance can be preserved by applying suitable numerical filtering at certain places of the discretization. While this filtering technique is effective for two-dimensional (2-D) periodic fluid interfaces, it does not apply to nonperiodic fluid interfaces. Moreover, using the filtering technique alone does not seem to be sufficient to stabilize 3-D fluid interfaces.

Here we introduce a new stabilizing technique for boundary integral methods for water waves which applies to nonperiodic and 3-D interfaces. A stabilizing term is added to the boundary integral method which exactly cancels the destabilizing term produced by the point vortex method approximation to the leading order. This modified boundary integral method still has the same order of accuracy as the point vortex method. A detailed stability analysis is presented for the point vortex method for 2-D water waves. The effect of various stabilizing terms is illustrated through careful numerical experiments.

     

Mixed multidimensional integral operators with piecewise constant kernels and their representations

We consider the algebra of mixed multidimensional integral operators. In particular, Fredholm integral operators belong to this algebra. For the piecewise constant kernels, we provide an explicit representation of the algebra as a direct product of simple matrix algebras. This representation allows us to compute the inverse operators and to find the spectrum explicitly. Moreover, explicit traces and determinants of such operators are also constructed. Generally speaking, the analysis of integral operators is reduced to the analysis of matrices.     

A new analytical technique to solve Volterra's integral equations

The purpose of the present paper is to show that the well‐known homotopy analysis method for solving ordinary and partial differential equations can be applied to solve linear and nonlinear integral equations of Volterra's type with high accuracy as well. Comparison of the present method with Adomian decomposition method (ADM), a well‐known method to solve integral equations, reveals that the ADM is only especial case of the present method. Furthermore, some illustrating examples such as linear, nonlinear and singular integral equations of Volterra's type are given to show high efficiency with reliable accuracy of the method. Copyright © 2011 John Wiley & Sons, Ltd.     

Interior Lp-estimates for functions with integral representations

For fnctions having either boundary integral representations or domain integral representations the present paper proves apriori estimate of first order derivatives in the L_pnorm. An overview of integral representations and their application to partial differential equations can be found in H. Begehr's and R. P.Gilbert's two volumes [2]. Interior esitmates in the Hijlder norm for very general (elliptic) systems of partial differential equations can be found in A. Douglis' and L. Nirenberg's paper [7], while estimates in the L_p-norm are contained, for instance, in the papers I ofthe threeauthors S. Agmon, A. Douglis and L. Nirenberg. In these papers [1,7] one can find further references.     

Application of Wiener-Hopf technique to linear nonhomogeneous integral equations for a new representation of Chandrasekhar's H-function in radiative transfer, its existence and uniqueness

In this paper, the linear nonhomogeneous integral equation of H-functions is considered to find a new form of H-function as its solution. The Wiener-Hopf technique is used to express a known function into two functions with different zones of analyticity. The linear nonhomogeneous integral equation is thereafter expressed into two different sets of functions having the different zones of regularity. The modified form of Liouville's theorem is thereafter used, Cauchy's integral formulae are used to determine functional representation over the cut region in a complex plane. The new form of H-function is derived both for conservative and nonconservative cases. The existence of solution of linear nonhomogeneous integral equations and its uniqueness are shown. For numerical calculation of this new H-function, a set of useful formulae are derived both for conservative and nonconservative cases.     

THE COMPOSITE FORMULAS ON A CONVEX DOMAIN IN Cn

Using the Plemelj formulas for a function and a (n, n − 1)-form on a convex domain in Cⁿ, the author obtains their composite formulas and inverse formulas. As an application, the author proves that the singular integral equation with Aizenberg kernel is equivalent to a Fredholm equation.     

A constructive integral equivalent to the integral of Kurzweil

We slightly modify the definition of the Kurzweil integral and prove that it still gives the same integral.