Calculus of Variations and Geometric Measure Theory
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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


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


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 diff erent behaviors at di fferent spatial scales, thus producing phenomena that are diff erent 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.


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