第二类典型域上的Cauchy型积分

本文在［１］的基础上，定义了第二类典型域上的Ｃａｕｃｈｙ主值，讨论了Ｃａｕｃｈｙ型积分，得到了一定条件下的边值的存在定理，给出了边界附近的值估计     

复双球垒域上奇异积分的Cauchy主值

利用一种新的挖法定义复双球垒域上的立体角系数,得到奇异积分的Cauchy主值的存在性.推广了复超球上的奇异积分理论.     

α次导数与Cauchy型积分

定义Laplace变换的像函数的任意实数次导数,同时给出了α次导数的性质,并建立了它和复围道积分的联系,给出了一类Cauchy型积分的计算公式.     

一类奇异积分和Cauchy型积分关于积分曲线的稳定性

本文讨论了当任意给定的f（τ,t）在某个区域E内属于H类时,奇异积分在封闭或开口光滑曲线E发生光滑扰动时的稳定性,并给出了相应的误差估计．作为应用,我们还讨论了当（t）在E内属于H类时,Cauchy型积分,在封闭光滑曲线E发生光滑扰动时的稳定性及误差估计．     

泛Clifford分析中无界域上的Cauchy积分公式和Cauchy-Pompeiu公式

本文研究了泛Clifford分析中的Cauchy积分公式和Cauchy-Pompeiu公式.通过引入修正的Cauchy核,得出了取值在泛Clifford代数上的两公式在无界域上的表达式.此两公式是有界域上的相应结果的推广,并为研究无界域上的边值问题打下了基础.     

积分型Cauchy中值定理的推广

介绍并证明积分型Cauchy中值定理的推广。     

第三类典型域的Busemann函数

本文利用第三类典型域测地线的矩阵形式算出其Busemann函数。     

积分型Cauchy中值定理的推广及逆问题

给出了一个积分型Cauchy中值定理的推广,并讨论了连续函数的积分型Cauchy中值定理的逆问题.     

积分型Cauchy中值定理中间点的渐近性

研究当积分区间长度趋于零或无穷时,积分型Cauchy积分中值定理中间点的渐近性质.     

带常系数的Cauchy型奇异积分方程的快速方法

Petrov-Galerkin 方法是研究Cauchy型奇异积分方程的最基本的数值方法. 用此方法离散积分方程可得一系数矩阵是稠密的线性方程组. 如果方程组的阶比较大, 则求解此方程组所需要的计算复杂度则会变得很大. 因此, 发展此类方程的快速数值算法就变成了必然. 该文将就对带常系数的Cauchy型奇异积分方程给出一种快速数值方法. 首先用一稀疏矩阵来代替稠密系数矩阵, 其次用数值积分公式离散上述方程组得到其完全离散的形式,然后用多层扩充方法求解此完全离散的线性方程组. 证明经过上述过程得到方程组的逼进解仍然保持了最优阶, 并且整个过程所需要的计算复杂度是拟线性的. 最后通过数值实验证明结论.     

A Note on Property of Cauchy Integral on Classical Domains

In[1],ZhengXueanprovedthat:letRI(n×n)befirstclassicaldomain,Unbecharacteris-ticboundaryofRI,x∈Un,f(x)∈L2(Un).Asz=rx(0≤r<1)→x,Cauchyintegral∫Unf(y)det(I-zy′)-ndyconvergetoafunctioninL2(Un).Inthispaper,wewillfacusourselfonCauchyintegralofL2onclassicaldomains[2]andgetsomeproperties.Themainresultisfollowing:Theorem　LetRbeoneofclassicaldomains,LbecharacteristicboundaryofR,f∈L2(L),H(z,ξ)beCauchykernelofRandF(z)=∫Lf(u)H(z,u)u　z∈R,(1)wheretherightisLebesgueintegral;uist…     

A Cauchy integral method of the solution of the 2D Dirichlet problem for simply or doubly connected domains

In this paper we propose a new method for solving the 2D Laplace equation with Dirichlet boundary conditions in simply and doubly connected domains. Here, we apply the numerical algorithm based on truncated Fourier series and reduce the corresponding Fredholm integral equation to a finite system of linear equations.     

A Hermitian Cauchy formula on a domain with fractal boundary

Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centered around the concept of monogenic functions, i.e. null solutions of a first order vector valued rotation invariant differential operator called Dirac operator, which factorizes the Laplacian; monogenic functions may thus also be seen as a generalization of holomorphic functions in the complex plane. Hermitian Clifford analysis offers yet a refinement of the Euclidean case; it focusses on the simultaneous null solutions, called Hermitian (or h-) monogenic functions, of two Hermitian Dirac operators which are invariant under the action of the unitary group. In Brackx et al. (2009) [8] a Clifford-Cauchy integral representation formula for h-monogenic functions has been established in the case of domains with smooth boundary, however the approach followed cannot be extended to the case where the boundary of the considered domain is fractal. At present, we investigate an alternative approach which will enable us to define in this case a Hermitian Cauchy integral over a fractal closed surface, leading to several types of integral representation formulae, including the Cauchy and Borel-Pompeiu representations.     

Principal values for the Cauchy integral and rectifiability

We give a geometric characterization of those positive finite measures on with the upper density finite at -almost every , such that the principal value of the Cauchy integral of ,

{\varepsilon}} \frac{1}{\xi-z}\, d\mu(\xi),\end{displaymath}">

exists for -almost all . This characterization is given in terms of the curvature of the measure . In particular, we get that for , -measurable (where is the Hausdorff -dimensional measure) with , if the principal value of the Cauchy integral of exists -almost everywhere in , then is rectifiable.

     

Boundedness and compactness characterizations of Cauchy integral commutators on Morrey spaces

Let C_Γ be the Cauchy integral operator on a Lipschitz curve Γ. In this article, the authors show that the commutator [b,C_Γ] is bounded (resp, compact) on the Morrey space for any (or some) p ∈ (1,∞) and λ ∈ (0,1) if and only if (resp, ). As an application, a factorization of the classical Hardy space in terms of C_Γ and its adjoint operator is obtained.     

A note on solving Cauchy integral equations of the first kind by iterations

The successive approximation method was applied for the first time by N.I. Ioakimidis to solve practical cases of a Cauchy singular integral equation: the airfoil one. In this paper we study a more general case. We prove the convergence of the method in this general case. The proposed method has been tested for two kernels which are particularly important in practice. Finally, some numerical examples illustrate the accuracy of the method.     

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.     

Cauchy Integral Formulae in Rn

In this note p((D)) =(D)m + b1(D)m-1+…+bm is a polynomial Dirac operator in Rn,where (D) =∑nj=1 ej ?(e)/(e)xj is a standard Dirac operator in Rn,bj are the complex constant coefficients.In this note we...     

Compact almost automorphic solutions to semilinear Cauchy problems with non-dense domain

In this work we study the existence and uniqueness of compact almost automorphic solutions to a first-order differential equation with a linear part dominated by a Hille-Yosida type operator with non dense domain.