**Abstract** : In geosciences and paleomagnetism, estimating the remanent magnetization in old rocks is an important issue to study past evolution of the Earth and other planets or bodies. However, the magnetization cannot be directly measured and only the magnetic field that it produces can be recorded.
In this paper we consider the case of thin samples, to be modeled as a planar set S of R^2 x {0}, carrying a magnetization m (a 3-dimensional vector field supported on S). This setup is typical of scanning microscopy that was developed recently to measure a single component of a weak magnetic field, close to the sample. Specifically, one is given a record of b_3[m] (tiny: a few nano Teslas), the vertical component of the magnetic field produced by m, on a planar region Q of R^2 x {h} located at some fixed height h > 0 above the sample plane. We assume that both S and Q are Lipschitz-smooth bounded connected open sets in their respective planes, and that the magnetization m belongs to [L^2(S)]^3, whence b_3[m] belongs to L^2(Q). Such magnetizations possess net moments <m> (belonging to R^3) defined as their integral on S.
Recovering the magnetization m or its net moment <m> from available measurements of b_3[m] are inverse problems for the Poisson-Laplace equation in the upper half-space R^3_+ with right hand side in divergence form. Indeed, Maxwell's equations in the quasi-static approximation identify the divergence of m with the Laplacian of a scalar magnetic potential in R^3_+ whose normal derivative on Q coincides with b_3[m]. Hence Neumann data b_3[m] are available on Q (subset of R^3_+), and we aim at recovering m or <m> on S. We thus face recovery issues on the boundary of the harmonicity domain from (partial) data available inside.
Such inverse problems are typically ill-posed and call for regularization. Indeed, magnetization recovery is not even unique, due to the existence of silent sources m != 0 such that b_3[m] = 0. And though such sources have vanishing moment so that net moment recovery is unique, estimation of the latter turns out to be unstable with respect to measurements errors.
The present work investigates silent sources, equivalent magnetization of minimal L^2(S)-norm to some given m in [L^2(S)]^3 (two magnetizations are called equivalent if their difference is silent), as well as density / instability results.