## S. Dipierro - M. Novaga - E. Valdinoci

# On a Minkowski geometric flow in the plane: Evolution of curves with lack of scale invariance

created by novaga on 15 Oct 2017

modified on 01 Feb 2019

[

BibTeX]

*Published Paper*

**Inserted:** 15 oct 2017

**Last Updated:** 1 feb 2019

**Journal:** J. London Math. Soc.

**Volume:** 99

**Number:** 1

**Pages:** 31-51

**Year:** 2019

**Abstract:**

We consider a planar geometric
flow in which the normal velocity is a
nonlocal variant of the curvature. The
flow is not scaling invariant and in fact
has different behaviors at different spatial scales, thus producing phenomena
that are different with respect to both the classical mean curvature
flow and
the fractional mean curvature
flow.
In particular, we give examples of neckpinch singularity formation, we
show that sets with "sufficiently small interior" remain convex under the
flow, but, on the other hand, in general the
flow does not preserve convexity.
We also take into account traveling waves for this geometric
flow, showing
that a new family of $C^2$ and convex traveling sets arises in this setting.

**Download:**