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Divergence-Free Measures in the Plane and Inverse Potential Problems in Divergence Form

Abstract : We show that a divergence-free measure on the plane is a continuous sum of unit tangent vector fields on rectifiable Jordan curves. This loop decomposition is more precise than the general decomposition in terms of elementary solenoids given by S.K. Smirnov when applied to the planar case. The proof involves extending the Fleming-Rishel formula to homogeneous BV functions (in any dimension), and establishing for such functions approximate continuity of measure theoretic connected components of suplevel sets as functions of the level. We apply these results to inverse potential problems whose source term is the divergence of some unknown (vector-valued) measure. A prototypical case is that of inverse magnetization problems when magnetizations are modeled by R 3-valued Borel measures. We investigate methods for recovering a magnetization µ by penalizing the measure theoretic total variation norm µ T V. In particular, we show that if a magnetization is supported in a plane, then T V-regularization schemes always have a unique minimizer, even in the presence of noise. It is further shown that T V-norm minimization (among magnetizations generating the same field) uniquely recovers planar magnetizations in the following cases: when the magnetization is carried by a collection of sufficiently separated line segments and a set that is purely 1-unrectifiable, or when a superset of the support is tree-like. We note that such magnetizations can be recovered via T V-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.
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https://hal.inria.fr/hal-02862882
Contributor : Laurent Baratchart <>
Submitted on : Tuesday, June 9, 2020 - 5:41:01 PM
Last modification on : Thursday, January 21, 2021 - 2:44:02 PM

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Laurent Baratchart, Cristobal Villalobos Guillén, Doug Hardin. Divergence-Free Measures in the Plane and Inverse Potential Problems in Divergence Form. 2020. ⟨hal-02862882⟩

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