Stationary Sets and Asymptotic Behavior of the Mean Curvature Flow with Forcing in the Plane

Abstract

We consider the flat flow solutions of the mean curvature equation with a forcing term in the plane. We prove that for every constant forcing term the stationary sets are given by a finite union of disks with equal radii and disjoint closures. On the other hand for every bounded forcing term tangent disks are never stationary. Finally in the case of an asymptotically constant forcing term we show that the only possible long time limit sets are given by disjoint unions of disks with equal radii and possibly tangent.

Stationary sets of the mean curvature flow with a forcing term

“…We also mention the work [7] where the authors study the same setting but add randomness to the flow. In our main theorem we generalize the result in [9] from the plane to the general case R n and prove that the flat flow instantaneously connects the two tangent balls with a thin neck which continues to grow at least for a short period of time.…”

“…Let us finally mention a few words about the proof of Theorem 1.1. We begin the proof as in the planar case [9] by showing that any discrete approximation of the flat flow creates at the first step a neck which connects the two balls. After this we need to show that this neck is growing until the time δ.…”

Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flow

“…Concerning the limiting configurations, Theorem 1.1 is sharp since the flow (1.1) may converge to tangent balls as it is shown in [14]. On the other hand, we believe that one can rule out the possible translations and the flow actually convergences to a disjoint union of balls.…”

“…The result in [19] shows that the convergence holds also for star-shaped sets, up to possible translations. For the mean curvature flow with forcing the asymptotic behavior has been studied for the level set solution in [15,16] and for the flat flow in the plane in [14]. The result closest to ours is the recent work by Morini-Ponsiglione-Spadaro [27], where the authors prove that the discrete-in-time approximation of the flat flow of (1.1) converges exponentially fast to disjoint union balls.…”

“…It would be interesting to obtain the sharp one as it might be crucial in order to obtain the possible exponential convergence of (1.1) as in [27]. In the two-dimensional case the optimal power q = 1 is proven in [14].…”