Divergence-free measures in the plane and inverse potential problems in divergence form

TL;DR

This paper characterizes divergence-free measures in the plane as sums of tangent vector fields on Jordan curves, extends key formulas to BV functions, and applies these results to inverse magnetization problems, demonstrating unique recovery via TV-regularization.

Contribution

It introduces a precise loop decomposition for divergence-free measures in the plane and applies it to inverse potential problems, establishing uniqueness and recovery conditions for planar magnetizations.

Findings

Divergence-free measures decompose into tangent vector fields on Jordan curves.

TV-regularization yields unique solutions for planar magnetizations.

Recovery is possible for magnetizations supported on separated line segments or tree-like sets.

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 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 R3-valued 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 TV-regularization schemes always have a unique minimizer, even in the presence of noise. It is further shown thatTV-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 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