Inverse potential problems in divergence form for measures in the plane

Baratchart, Laurent; Villalobos Guillén, Cristóbal; Hardin, Douglas P.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 87

We study inverse potential problems with source term the divergence of some unknown (ℝ³-valued) measure supported in a plane; e.g., inverse magnetization problems for thin plates. We investigate methods for recovering a magnetization $$ by penalizing the measure-theoretic total variation norm ∥$$∥$$, and appealing to the decomposition of divergence-free measures in the plane as superpositions of unit tangent vector fields on rectifiable Jordan curves. In particular, we prove for magnetizations supported in a plane that TV -regularization schemes always have a unique minimizer, even in the presence of noise. It is further shown that TV -norm minimization (among magnetizations generating the same field) uniquely recovers planar magnetizations in the following two cases: (i) when the magnetization is carried by a collection of sufficiently separated line segments and a set that is purely 1-unrectifiable; (ii) when a superset of the support is tree-like. We note that such magnetizations can be recovered via TV -regularization schemes in the zero noise limit by taking the regularization parameter to zero. This suggests definitions of sparsity in the present infinite dimensional context, that generate results akin to compressed sensing.

Classification : 31B20, 49N45, 49Q20, 86A22
Keywords: Planar divergence free measures, purely 1-unrectifiable sets, inverse potential problems in divergence form, thin plate magnetizations, Sparse recovery, total variation regularization

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