Dimensional estimates and rectifiability for measures satisfying linear PDE constraints

By Adolfo Arroyo-Rabasa, Guido De Philippis, J. Hirsch, Filip Rindler. Geometric and Functional Analysis, 2018

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We establish the rectifiability of measures satisfying a linear PDE constraint. The obtained rectifiability dimensions are optimal for many usual PDE operators, including all first-order systems and all second-order scalar operators. In particular, our general theorem provides a new proof of the rectifiability results for functions of bounded variations (BV) and functions of bounded deformation (BD). For divergence-free tensors we obtain refinements and new proofs of several known results on the rectifiability of varifolds and defect measures.

Subfields: Applied Mathematics, Biomedical Engineering, Numerical Analysis

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