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Stationary measures and rectifiability

Authors:

Email author" target="_blank">Roger?Moser Email author

Institution:

(1) MPI for Mathematics in the Sciences, Inselstr. 22-26, 04103 Leipzig, Germany

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

$\int_\Omega \left({\rm div} \phi d\mu_{\alpha\alpha} - p \, \frac{\partial\phi^\alpha}{\partial x^\beta} d\mu_{\alpha\beta}\right) = 0$

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.Received: 4 April 2002, Accepted: 16 June 2002, Published online: 5 September 2002Mathematics Subject Classification (1991): 49Q15, 49Q05, 58E20, 58E15

Keywords:

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