@InProceedings{BCL2017,
  author = 	 {Baratchart, Laurent and Chevillard, Sylvain and Leblond, Juliette},
  title = 	 {{Silent and equivalent magnetic distributions on thin plates}},
  booktitle = {{Harmonic Analysis, Function Theory, Operator Theory, and their Applications}},
  year = 	 {2017},
  editor = 	 {Jaming, Philippe and Hartmann, Andreas and Kellay, Karim and Kupin, Stanislas and Pisier, Gilles and Timotin, Dan},
  series = 	 {{Theta Series in Advanced Mathematics}},
  pages = 	 {11--27},
  publisher = {{The Theta Foundation}},
  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 $S \subset \RR^2 \times \{0\}$, carrying
		a magnetization $\bm$ (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 $b_3[\bm]$ (tiny: a few nano Teslas), the vertical
		component of the magnetic field produced by $\bm$, on a planar
		region $Q\subset \RR^2 \times \{h\}$ 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 $\bm$
		belongs to $[L^2(S)]^3$, whence $b_3[\bm] \in L^2(Q)$. Such
		magnetizations possess net moments
		$\langle \bm \rangle \in \RR^3$ defined as their integral on
		$S$.

                Recovering the magnetization $\bm$ or its net moment
		$\langle \bm \rangle $ from available measurements of $b_3[\bm]$
		are inverse problems for the Poisson-Laplace equation in the
		upper half-space $\RR^3_+$ with right hand side in divergence
		form. Indeed, Maxwell's equations in the quasi-static
		approximation identify the divergence of $\bm$ with the
		Laplacian of a scalar magnetic potential in $\RR^3_+$ whose
		normal derivative on $Q$ coincides with $b_3[\bm]$. Hence
		Neumann data $b_3[\bm]$ are available on $Q \subset \RR^3_+$,
		and we aim at recovering $\bm$ or $\langle \bm \rangle$ 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 $\bm \neq 0$ such
		that $b_3[\bm] = 0$. 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
		$\bm \in [L^2(S)]^3$ (two magnetizations are called equivalent
		if their difference is silent), as well as density / instability
		results.
             },
  keywords = {complex rational approximation, Hardy spaces, lower bounds, error rates}
}