Stationary measures and rectifiability

Roger Moser
August 2003
Calculus of Variations & Partial Differential Equations;Aug2003, Vol. 17 Issue 4, p357
Academic Journal
For integers $1 \le p < n$, we consider $\mathbb{R}�{n \times n}$-valued Radon measures $\mu = (\mu_{\alpha\beta})$ on an open set $\Omega \subset \mathbb{R}�n$which satisfy $$\int_\Omega \left({\rm div} \phi d\mu_{\alpha\alpha} - p \, \frac{\partial\phi�\alpha}{\partial x�\beta} d\mu_{\alpha\beta}\right) = 0$$ for all $\phi \in C_0�1(\Omega,\mathbb{R}�n)$. We show that under certain conditions, $\mu$]*> has an ( n - p)-dimensional density everywhere, and the set of points of positive density is countably ( n - p)-rectifiable. This simplifies the proofs of several rectifiability theorems involving varifolds with vanishing first variations, p-harmonic maps, or Yang-Mills connections.


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