Affiliation:
1. Department of Mathematics and Statistics , University of Jyvaskyla , Jyvaskyla , Finland
Abstract
Abstract
We consider the flat flow solution to the mean curvature equation with forcing in
ℝ
n
{\mathbb{R}^{n}}
.
Our main result states that tangential balls in
ℝ
n
{\mathbb{R}^{n}}
under a flat flow with a bounded forcing term will
experience fattening, which generalizes the result in
[N. Fusco, V. Julin and M. Morini,
Stationary sets and asymptotic behavior of the mean curvature flow with forcing in the plane,
preprint 2020, https://arxiv.org/abs/2004.07734]
from the planar case to higher dimensions. Then, as in the planar case, we characterize stationary sets in
ℝ
n
{\mathbb{R}^{n}}
for a constant forcing term as finite unions of equisize balls with mutually positive distance.
Subject
Applied Mathematics,Analysis
Reference17 articles.
1. F. Almgren, J. E. Taylor and L. Wang,
Curvature-driven flows: A variational approach,
SIAM J. Control Optim. 31 (1993), no. 2, 387–438.
2. M. Barchiesi, F. Cagnetti and N. Fusco,
Stability of the Steiner symmetrization of convex sets,
J. Eur. Math. Soc. (JEMS) 15 (2013), no. 4, 1245–1278.
3. G. Bellettini,
Lecture Notes on Mean Curvature Flow, Barriers and Singular Perturbations,
Appunti. Sc. Norm. Super. Pisa (N. S.) 12,
Edizioni della Normale, Pisa, 2013.
4. G. Bellettini and M. Paolini, Some results on minimal barriers in the sense of De Giorgi applied to driven motion by mean curvature, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 19 (1995), 43-67
5. Erratum, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 26 (2002), 161-165.