On obstacle problem for mean curvature flow with driving force

Abstract

In this paper, we study an obstacle problem associated with the mean curvature flow with constant driving force. Our first main result concerns interior and boundary regularity of the solution. We then study in details the large time behavior of the solution and obtain the convergence result. In particular, we give full characterization of the limiting profiles in the radially symmetric setting.2010 Mathematics Subject Classification. 35A01, 35A02, 35K55, 53C44.

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

“…We remark that the same type of classification holds true in the framework of level-set solutions, as recently shown in [16,Theorem 4.7]. The general n-dimensional case remains open also for the viscosity solutions, see [17]. 1.3.…”

“…We also mention [23] where the long-time behavior of the discrete Euler implicit scheme for the volume preserving mean curvature flow is addressed for any arbitrary bounded initial set with finite perimeter. The long time behavior of the forced mean curvature flow in the context of viscosity level-set solutions was also investigated in [17] and [16] where it is shown that under certain assumptions the solutions converge to a stationary solution of the level-set equation. The problem of classifying the latter is open in general.…”

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

“…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.…”

Remarks on the generalized Cauchy-Dirichlet problem for graph mean curvature flow with driving force

“…In [13], global time existence and uniqueness of C 1;1 solutions to the mean curvature flow with obstacles in a periodic setting is established, and the long time behavior is studied as well. In [10] the obstacle problem for the level set equation of forced mean curvature flows is considered, and the large time behavior of the solution is studied. The study of this paper is inspired by the work [4] which studies traveling waves for reaction diffusion equations with obstacles.…”