Inverse potential problems in divergence form for measures in the plane

We study inverse potential problems with source term the divergence of some unknown (ℝ 3 -valued) measure supported in a plane; e.g. , magnetization for thin plates. investigate methods recovering μ by penalizing measure-theoretic total variation norm ∥ TV and appealing to decomposition divergence-free measures plane as superpositions unit tangent vector fields on rectifiable Jordan curves. In particular, we prove magnetizations that -regularization schemes always have unique minimizer, even presence noise. It is further shown -norm minimization (among generating same field) uniquely recovers planar following two cases: (i) when carried collection sufficiently separated line segments set purely 1-unrectifiable; (ii) superset support tree-like. note such can be recovered via zero noise limit taking regularization parameter zero. This suggests definitions sparsity present infinite dimensional context, generate results akin compressed sensing.