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)

Abstract

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).

Keywords

Assure Nash 

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References

  1. [H]
    R.S. Hamilton, “The inverse function theorem of Nash and Moser”, Bull Amer. Math. Soc. (1982).Google Scholar
  2. [HÖ]
    L. Hörmander, “The boundary problems of physical geodesy”, Arch. Rat. Mech. Anal. (1976).Google Scholar
  3. [MPS]
    C. Maderna, C. Pagani and S. Salsa, “Nonlinear analysis PMA. Vol. 10, 1986.Google Scholar
  4. [P]
    C.D. Pagani “Stability of a surface determined from measures of potential”, to appear in SIAM.Google Scholar
  5. [S]
    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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