Analytic solutions to the inverse problem of the Newtonian potential (Q1112980)

The paper deals with a classical inverse problem which is unsolved in its general form. The figure of a homogeneous material body G is to be determined by measurements of the Newtonian potential on the surface of a sphere containing G in its interior. The authors assume that \(\Gamma =\partial G\) is diffeomorphic to \(S^ 2=\{x\in R^ 3:\quad x=1\},\) u: \(S^ 2\mapsto R\) satisfies \(| u| \leq c<1\). Further, let \(\phi_ u: S^ 2\mapsto R^ 3\) be the differential embedding defined by \[ \phi_ u(\omega)=\omega (1+u(\omega)),\quad \omega \in S^ 2,\quad \Gamma_ u=\phi_ u(S^ 2). \] \(G_ u\) denotes the domain whose boundary is \(\Gamma_ u\). Consider the potential created by a mass of unit density distributed over \(G_ u:\) \[ V_ u(x)=\int_{S^ 2}d\omega \int^{| \phi_ u(\omega)|}_{0} | x-t\omega |^{-1} t^ 2 dt. \] This potential is measured on \(\Gamma_ 1\) (i.e. \(| x| =2)\). Let v be the difference between this measured potential and the potential created by \(G_ 0\) at \(\Gamma_ 1\) (i.e. \(\pi\) /3). The aim is to prove an existence theorem for the equation \(B(u)=v\), where \(B(u)(\omega)=V_ u(2\omega)-\pi /3\). Here v is given and authors want to solve this equation for u. The method used is a variant of the Nash-Moser iteration technique. In the paper several hard Lemmas are proved which are of special interest.