Inverse Gravimetry Problem

where A is some set of multiindices α. A first crucial question is whether there is enough data to (uniquely) find f . Let u0 be a function in the Sobolev space H(Ω) with zero Cauchy data u0 = ∂νu0 = 0 on Γ0 and let f0 = −∆u0. Due to linearity, −∆(u+ u0) = f + f0. Obviously, u and u+ u0 have the same Cauchy data on Γ0, so f and f+f0 produce the same data (2), but they are different in Ω. It is clear, that there is a very large (infinite dimensional) manifold of solutions to the inverse problems of gravimetry. To regain uniqueness one has to restrict unknown distributions to a smaller but physically meaningful uniqueness class. Since f with the data (2) is not unique one can look for f with the smallest (L(Ω)-) norm. The subspace of harmonic functions fh is L -closed, so for any f there is a unique fh