Abstract

In this paper, we investigate the inverse problem of a simple layer potential where, by the known values of the potential gradient given as a finite series in a neighborhood of the point at infinity, we search for a simple closed contour with gravitating masses that generates potential outside the contour. The desired contour is an image of the unit circle under a mapping by polynomials of a special kind that are integral representations of another polynomials.

The main results of the paper are given in three theorems. In Theorem 1, for such representations that depend on two parameters, we obtain a demonstrative convexity criterion as a bounded domain on the complex plane: belonging of a pair of these coefficients as a point of the plane to this domain is equivalent to the convexity. Estimates and their asymptotic behavior are obtained for the star-likeness of the specified representations. The central result of the paper is Theorem 2 which shows that the inverse problem of a simple layer potential, resolvable in the form of a polynomial, cannot have two convex solutions. In Theorem 3, we give a simple sufficient univalence condition (connected to the Noshiro–Varshavsky criterion) for such polynomials. We also present the results on close-to-convexity of the given integral representations.

中文翻译：

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简单层势反问题中积分表示的几何性质

摘要

在本文中，我们研究了简单层势的逆问题，其中，通过在无穷远点附近作为有限级数给出的势梯度的已知值，我们搜索具有引力质量的简单闭合轮廓，产生轮廓外的电位。所需的轮廓是单位圆在特殊类型多项式映射下的图像，这些多项式是另一个多项式的整数表示。

论文的主要结果在三个定理中给出。在定理 1 中，对于这种依赖于两个参数的表示，我们获得了一个证明凸性标准作为复平面上的有界域：这些系数中的一对作为平面上的一个点属于该域相当于凸性。估计值及其渐近行为是针对指定表示的星形相似性获得的。该论文的核心结果是定理 2，它表明可以以多项式的形式解决的简单层势的逆问题不能有两个凸解。在定理 3 中，我们为这些多项式给出了一个简单的充分单价条件（与 Noshiro-Varshavsky 准则相关）。我们还展示了给定积分表示的接近凸性的结果。