Calculus of Variations and Geometric Measure Theory

G. Bellettini - S. Kholmatov

Minimizing movements for the generalized power mean curvature flow

created by kholmatov2 on 29 Mar 2024
modified on 03 Apr 2024



Inserted: 29 mar 2024
Last Updated: 3 apr 2024

Year: 2024


Motivated by a conjecture of De Giorgi, we consider the Almgren-Taylor-Wang scheme for mean curvature flow, where the volume penalization is replaced by a term of the form \[ \int_{E\Delta F} f\Big(\frac{ {\rm d}_F }{\tau}\Big)~dx \] for $f$ ranging in a large class of strictly increasing continuous functions. In particular, our analysis covers the case \[ f(r) = r^\alpha, \qquad r \geq 0, \quad \alpha>0, \] considered by De Giorgi. We show that the generalized minimizing movement scheme converges to the geometric evolution equation \[ f(v) = - \kappa\quad \text{on $\partial E(t)$,} \] where $\{E(t)\}$ are evolving subsets of $\mathbb{R}^n,$ $v$ is the normal velocity of $\partial E(t),$ and $\kappa$ is the mean curvature of $\partial E(t)$. We extend our analysis to the anisotropic setting, and in the presence of a driving force. We also show that minimizing movements coincide with the smooth classical solution as long as the latter exists. Finally, we prove that in the absence of forcing, mean convexity and convexity are preserved by the weak flow.

Tags: GeMeThNES
, GeoMeG