Geometric Properties of Integral Representations in the Inverse Problem of a Simple Layer Potential

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.