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Mathematics > Analysis of PDEs

arXiv:1812.07629 (math)
[Submitted on 18 Dec 2018]

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

Authors:Adolfo Arroyo-Rabasa
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Abstract: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.
Comments: 10 pages
Subjects: Analysis of PDEs (math.AP)
MSC classes: Primary 28A78, 49Q15, Secondary 35F35
Cite as: arXiv:1812.07629 [math.AP]
  (or arXiv:1812.07629v1 [math.AP] for this version)
  https://doi.org/10.48550/arXiv.1812.07629

Submission history

From: Adolfo Arroyo-Rabasa [view email]
[v1] Tue, 18 Dec 2018 20:29:31 UTC (21 KB)
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