The Nash-Moser Technique for an Inverse Problem in Potential Theory Related to Geodesy

  • C. Maderna
  • C. Pagani
  • S. Salsa
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 3)


We shall consider the following inverse problem of potential theory: to determine the shape of a body from measurements of the Newtonian potential at its surface, given some information about its distribution. More precisely, we consider a class of bodies G that can be parametrized as follows. Let u be a smooth mapping of S2 → R with |Cu| ⩽ constant < r0. (Here S2 is the surface of the unit ball in R3). For w e S2, define
$$ {\phi_{\text{u}}}{\text{(w) = }}{{\text{r}}_0} + {\text{u(w),}}\,{{\text{r}}_{\text{u}}}{ = }{\phi_0}({{\text{S}}^2}) $$
Then G ≡ GU is the bounded domain whose boundary is rU. (Thus r0 is reference sphere with center at the origin and radius r0).


Inverse Problem Spherical Harmonic Radial Function Fredholm Integral Equation Newtonian Potential 
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    C. Maderna, C. Pagani and S. Salsa, “Nonlinear analysis PMA. Vol. 10, 1986.Google Scholar
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    C.D. Pagani “Stability of a surface determined from measures of potential”, to appear in SIAM.Google Scholar
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    D.G. Schaeffer, “The capacitor problem”, Indiana Univ. Math. J. (1975).Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • C. Maderna
    • 1
  • C. Pagani
    • 2
  • S. Salsa
    • 2
  1. 1.Dipartimento di MatematicaFederigo EnriquesMilanoItaly
  2. 2.Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly

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