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MiS Preprint
29/2002
Stationary measures and rectifiability
Roger Moser
Abstract
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 $$ \int_\Omega \left(div \, \phi \, d\mu_{\alpha\alpha} - p \, \frac{\partial\phi^\alpha}{\partial x^\beta} \, d\mu_{\alpha\beta}\right) = 0 $$ 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.
