On the determination of a measure by the orbits generated by its logarithmic potential

Abstract

Abstract. We say that a logarithmic potential generates a curve in the plane if a unit mass traces the curve under the action of the potential. We consider the following problem: A one-parameter family of plane curves is given. We assume that these curves lie in the complement of a compact set K. Find all measures supported in K whose potentials generate each of the given curves. We solve this problem when K is the unit circle in three specific cases: (a) when the given curves are straight lines through the origin, (b) when the curves are straight lines through a point on the unit circle, and (c) when the curves are circles centered at the origin. The solution involves the Poisson integral and its boundary behavior. An inverse problem in potential theoryLet σ be a finite Borel measure with support in a compact set K that lies in the plane R 2 . For notational convenience we identify R 2 with the complex plane C. The logarithmic potential of σ is the function V σ :It is well known (see e.g. [6, Ch.3]) that V σ is a function superharmonic in C and harmonic in the complement of its support (which contains the set C \ K). Let α be an orbit in the plane. By orbit here we mean a C 2 function α(t) = x(t) + iy(t) : I → C defined on a closed interval I ⊂ R. A curve is a set of points x + iy ∈ C that satisfy an equation of the form f(x, y) = q, where q is a real number and f is a C 2 real function defined on an open subset of C. By the usual abuse of language, we will say that an orbit α lies in a set A ⊂ C if α(t) ∈ A for all t ∈ I. Of course every orbit α lies on the curve {α(t) : t ∈ I} which is the trace of α. The equation f(x, y) = q of the trace can be found (in principle) by eliminating t from the equations x = x(t), y = y(t).We say that the potential V σ generates the orbit α if Newton's second law holds:(