论文标题
通用奇异椭圆方程的双层和简单的潜力及其在解决dirichlet问题上的应用
Double- and simple-layer potentials for generalized singular elliptic equations and their applications to the solving the Dirichlet problem
论文作者
论文摘要
电位在解决椭圆方程的边界价值问题中起着重要作用。在上个世纪中叶，为具有一个单数系数的二维椭圆方程构建了潜在的理论。在对电势的研究中，给定方程的基本解决方案的特性本质上是有效的。目前，已经知道具有多个奇异系数的多维椭圆方程的基本解决方案。在本文中，我们研究了这种椭圆方程的双层和简单势。潜在理论的结果使我们能够代表积分方程形式中边界价值问题的解决方案。通过在许多变量中使用Lauricella的超几何函数的分解公式和其他身份，我们证明了限制定理并得出有关双重和简单势的密度的积分方程。将获得的结果应用于在多维球的某些部分中找到通用的单数椭圆方程的Dirichlet问题的明确解决方案。
Potentials play an important role in solving boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the study of potentials, the properties of the fundamental solutions of the given equation are essentially and fruitfully used. At the present time, fundamental solutions of a multidimensional elliptic equation with several singular coefficients are already known. In this paper, we investigate the double- and simple-layer potentials for this kind of elliptic equations. Results from potential theory allow us to represent the solution of the boundary value problems in integral equation form. By using a decomposition formula and other identities for the Lauricella's hypergeometric function in many variables, we prove limiting theorems and derive integral equations concerning a densities of the double- and simple-layer potentials. The obtained results are applied to find an explicit solution of the Dirichlet problem for the generalized singular elliptic equation in the some part of the multidimensional ball.
