On a minimal element for a family of bodies producing the same external gravitational field

Abstract

A minimal element, mother body, for a family of bodies producing the same external gravitational field is the body in the family, whose support has Lebesgue measure zero and satisfies some additional requirements. The finite algorithm of constructing mother bodies in

is suggested. The local structure of mother bodies near singular points of continued logarithmic potential is investigated in generic positions.

Keywords:

• Analytic continuation
• Mother body
• Logarithmic potential
• Balayage
• Schwarz function
• AMS Subject Classifications: 32Dxx
• 31Axx

Acknowledgments

We are grateful to Harold Shapiro, who attracted our attention to this problem and to Björn Gustafsson who had acquainted us with the preprint of his interesting article [9].

Notes

From now on we do not distinguish the function V(x, y) and its continuation

as well as functions of (x, y) and functions of (z, ζ).

The derivative

plays a crucial role in the investigation of singularities of continuation of potential

inside the domain

. This derivative is known in the Russian geophysical literature as a complex vector of gravitational field (see, e.g., [6]).