Dear Matt: The people who are actively working with this whom I know are [Genady Lyubeznik](http://www.math.umn.edu/~gennady/) and [Manuel Blickle](http://www.mabli.org/mabli.html). 
The key point seems to be that certain $R[F]$-modules become simpler when viewed as $D_R$-modules (here $F$ is the Frobenius). It has been applied to show that certain local cohomology modules over regular local rings in positive characteristic have finitely many associated primes. Examples are in the following papers:


* _[D-module structure of R[F]-modules](https://arxiv.org/abs/math/0201180)_, Trans. Am. Math. Soc **355** (2003), 4, 1647–1668, and 

* _[Generators of D-modules in positive characteristic](https://arxiv.org/abs/math/0502405)_, Math. Res. Lett. **12** (2005), no. 4, 459–473,

and the references therein. There is also this new preprint, Lyubeznik, Zhang and Zhang, _[A Property of the Frobenius Map of a Polynomial Ring](https://arxiv.org/abs/1001.2949)_, which might be of interest.