Divergencefree measures in the plane and inverse potential problems in divergence form
Abstract
We show that a divergencefree 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 elementary solenoids given by S.K. Smirnov, when applied to the planar case. The proof involves extending the FlemingRishel 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 (vectorvalued) measure. A prototypical case is that of inverse magnetization problems when magnetizations are modeled by R3valued Borel measures. We investigate methods for recovering a magnetization {\mu} by penalizing its measure theoretic total variation norm (TV). In particular, we prove that if a magnetization is supported in a plane, then TVregularization schemes always have a unique minimizer, even in the presence of noise. It is further shown thatTVnorm 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 1unrectifiable, or when a superset of the support is treelike. We note that such magnetizations can be recovered via TVregularization 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
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.09072
 Bibcode:
 2020arXiv200609072B
 Keywords:

 Mathematics  Functional Analysis;
 46 N;
 46 G (Primary);
 49 Q (Secondary)
 EPrint:
 40 pages