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On the Newtonian potential of ellipsoids. (English) Zbl 0830.31003
It is known that the Newtonian potential of a uniform mass distribution of an ellipsoid is equal to a quadratic polynomial inside the ellipsoid. In 1931 P. Dive [Bull. Soc. Math. Fr. 59, 128-140 (1931; Zbl 0004.16601)] proved that the converse is valid – if $$K$$ is a bounded solid in $$\mathbb{R}^3$$ and its Newtonian potential is equal to a quadratic polynomial inside it, then $$K$$ is an ellipsoid; in 1986 E. DiBenedetto and A. Friedman [Indiana Univ. Math. J. 35, 573-606 (1986; Zbl 0667.35074)] generalized this result to the case of $$\mathbb{R}^m$$, $$m > 2$$.
The author uses some topological methods to obtain a simpler proof of that result.
Reviewer: M.Dont (Praha)

##### MSC:
 31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
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