Calculus of Variations and Geometric Measure Theory

A. Arroyo-Rabasa

An elementary approach to the dimension of measures satisfying a first-order linear PDE constraint

created by arroyorabasa on 25 Jun 2019


Accepted Paper

Inserted: 25 jun 2019
Last Updated: 25 jun 2019

Journal: Proceedings of the American Mathematical Society
Year: 2018

ArXiv: 1812.07629 PDF

10 pages


We give a simple criterion on the set of probability tangent measures $\mathrm{Tan}(\mu,x)$ of a positive Radon measure $\mu$, which yields lower bounds on the Hausdorff dimension of $\mu$. As an application, we give an elementary and purely algebraic proof of the sharp Hausdorff dimension lower bounds for first-order linear PDE-constrained measures; bounds for closed (measure) differential forms and normal currents are further discussed. A weak structure theorem in the spirit of Ann. Math. 184(3) (2016), pp. 1017-1039 is also discussed for such measures.

Keywords: Hausdorff dimension, Normal current, $\mathcal A$-free measure, structure theorem, PDE constraint, tangent measure