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On the regular decomposition of the inverse gravimetric problem in non-L 2 spaces F. Sansò

Received: 9 May 2013 / Accepted: 12 January 2014 © Springer-Verlag Berlin Heidelberg 2014

Abstract The paper deals with the inverse gravimetric problem generalizing a classical decomposition of mass distributions into a harmonic component and another component that produces a zero external field. After a review and an extension of the well-known L 2 theory, mass distributions in L p and in H −s,2 are considered and proved to undergo an analogous decomposition. Examples will make the theory easier to grasp. Conclusions follow. Keywords

Gravity field · Inverse problems · Regular decomposition

Mathematics Subject Classification

31B20

1 An introduction to the inverse gravimetric problem Assume we have a body B in R 3 , with B = B ∪ S, B the interior of B and S its boundary. We shall agree B to be compact and simply connected, and we will denote by  the exterior of B; so  is an (unbounded) open set, with boundary S too. To be specific, we shall assume from now on that S is at least a Lipschitz surface (see Miranda 1970; McLean 2000), introducing later on more restrictive conditions. Assume now that f is some mass distribution on B. For the moment we can think of f as a measure of finite variation with support in B or even as a measure with a measurable density f (x)(x ∈ B). If we call N (x) =

1 |x|−1 4π

(1)

F. Sansò (B) Politecnico di Milano, DICA, Polo Territoriale di Como, via Valleggio, 11, 22100 Como, Italy e-mail: [email protected]

123

Int J Geomath

the Newton kernel, then the external Newtonian potential of f is x ∈ , u(x) =< N (x − y), f >≡ N ( f ),

(2)

where the coupling (2) can be thought of as an integral over the measure f , or, when it exists, as the Lebesgue integral on B of N (x − y) f (y), Remark 1 The theory developed in this paper has an obvious generalization to R n , n > 3; nevertheless since we like to frame our problem within the geophysical or geodetic point of view, thinking of u(x) as the exterior gravity potential of the Earth, we will carry out our proofs in R 3 . The inverse gravimetric problem is to determine f from u, i.e. to find the mass source f generating the external potential u. As a matter of fact this is one of the oldest improperly posed problems treated in the literature on inverse problems (Lavrentiev 1967; Tichonov and Arsenin 1977); probably first met in 1867 in a paper by Stokes (1867). From the mathematical point of view the interesting feature comes from the large indetermination of the solution of (2), due to the existence of a wide class of functions (mass densities) f that produce a zero outer potential, i.e. the Newton operator N (·) has a large null space. This effect has already been highlighted in a number of papers in the 19th century and at the beginning of the 20th century. To better appreciate it let us start with an obvious proposition, (see for instance Gilbarg and Trudinger 1983, §2.4), that will be useful in the sequel too. We use the standard not