TY - UNPB

T1 - The nonlocal mean curvature flow of periodic graphs

AU - Matioc, Bogdan-Vasile

AU - Walker, Christoph

N1 - 32 pages

PY - 2022/7/15

Y1 - 2022/7/15

N2 - We establish the well-posedness of the nonlocal mean curvature flow of order \({\alpha\in(0,1)}\) for periodic graphs on \(\mathbb{R}^n\) in all subcritical little H\"older spaces \({\rm h}^{1+\beta}(\mathbb{T}^n)\) with \(\beta\in(0,1)\). Furthermore, we prove that if the solution is initially sufficiently close to its integral mean in \({\rm h}^{1+\beta}(\mathbb{T}^n)\), then it exists globally in time and converges exponentially fast towards a constant. The proofs rely on the reformulation of the equation as a quasilinear evolution problem, which is shown to be of parabolic type by a direct localization approach, and on abstract parabolic theories for such problems.

AB - We establish the well-posedness of the nonlocal mean curvature flow of order \({\alpha\in(0,1)}\) for periodic graphs on \(\mathbb{R}^n\) in all subcritical little H\"older spaces \({\rm h}^{1+\beta}(\mathbb{T}^n)\) with \(\beta\in(0,1)\). Furthermore, we prove that if the solution is initially sufficiently close to its integral mean in \({\rm h}^{1+\beta}(\mathbb{T}^n)\), then it exists globally in time and converges exponentially fast towards a constant. The proofs rely on the reformulation of the equation as a quasilinear evolution problem, which is shown to be of parabolic type by a direct localization approach, and on abstract parabolic theories for such problems.

KW - math.AP

KW - 35K59, 35K93, 35B35, 35B65

U2 - 10.48550/arXiv.2207.07474

DO - 10.48550/arXiv.2207.07474

M3 - Preprint

BT - The nonlocal mean curvature flow of periodic graphs

ER -