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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

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.

Received:
21.03.02
Published:
21.03.02
MSC Codes:
49Q15, 49Q05, 58E20, 58E15
Keywords:
rectifiability, minimal varifolds, p-harmonic maps, yang-mills connections

Related publications

inJournal
2003 Repository Open Access
Roger Moser

Stationary measures and rectifiability

In: Calculus of variations and partial differential equations, 17 (2003) 4, pp. 357-368