论文标题
通过潜在方法解决三维椭圆方程的Dirichlet和Holmgren问题
Solving the Dirichlet and Holmgren problems for a three-dimensional elliptic equation by the potential method
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，已经知道了具有一个变性线的三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。通过使用高斯高几何函数的某些属性，我们证明了限制定理并得出有关双层和简单势势的密度的积分方程。将获得的结果应用于找到球的三维奇异椭圆方程的Dirichlet和Holmgren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 three-dimensional elliptic equation with one degeneration line 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 some properties of Gaussian hypergeometric function, 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 and Holmgren problems for the three-dimensional singular elliptic equation in the half of the ball.
