闭光滑流形上的高阶线性微-积分方程

The Higher Order Linear Differential-Integral Equations on Smooth Closed Orientable Manifolds

利用积分变换技巧,作者给出了C^n中闭光滑可定向流形上一个新的Bochner-Martinelli型积分的高阶偏导数的奇异积分的Hadamard主值,获得了高阶奇异积分的Plemelj公式和合成公式,还讨论了相应的变系数线性微分积分方程的正则化,证明其可转化为一类等价的Fredholm方程。并且指出其特征方程当给出一组适当的边值条件时,在L~*中存在唯一解。

By using the technique of integral transformation, the authors give a new definition of Hadamard principal value of singular integrals of higher order partial derivatives for the integral of Bochner-Martinelli type on a closed smooth orientable manifold in C^n. The Plemelj formula and compsite formula of higher order singular integral are obtained. The regularization of differential singular integral equations with linear variable coefficients is discussed. The differential singular integral equation can be transformed into an equivalent Fredholm equation, whose characteristic equation has an unique solution in L~* under suitable boundary value conditions.