Level-set forced mean curvature flow with the Neumann boundary condition

Abstract

Here, we study a level-set forced mean curvature flow with the homogeneous Neumann boundary condition. We first show that the solution is Lipschitz in time and locally Lipschitz in space. Then, under an additional condition on the forcing term, we prove that the solution is globally Lipschitz. We obtain the large time behavior of the solution in this setting and study the large time profile in some specific situations. Finally, we give two examples demonstrating that the additional condition on the forcing term is sharp, and without it, the solution might not be globally Lipschitz.

Capillary-type boundary value problems of mean curvature flow with force and transport terms on a bounded domain

“…Note that the illustration justifies the necessity of a nonzero force term in order to have a global gradient estimate on a nonconvex domain. In this context, the results on the forced mean curvature flow with the right angle condition have been obtained [23] recently, which explains the effect of the constraints by the forcing term and by the geometry of the boundary. However, there are no results on the forced mean curvature flow and the forced mean curvature equation with more general boundary conditions on a general bounded domain, for neither the graph case nor the level-set case.…”

“…The novelty of this paper is threefold; first of all, the multiplier method in [23] can be combined with the method in [40] in order to get a priori gradient estimates of (1.7) uniform in η > 0. The combination of the methods allows us to handle the difficulties coming from the nonconvexity of Ω, a forcing term c, a transport term f , a nonzero boundary condition with φ ≡ 0 at the same time.…”

“…We accordingly are able to study the mean curvature equations (1.5) and (1.6). Finally, by adopting the approaches in [16,23], we discuss the optimality of the coercive condition on c, and compute the eigenvalue, the large time profile based on the optimal control formula in the radially symmetric setting of (1.2). We also give a dynamics proof in order to deal with the boundary, which does not appear in [16], when we study the asymptotic behavior.…”

“…The multiplier method in [23] has been considered new and devised only recently, and it successfully treats the homogeneous Neumann boundary condition. The method is natural, and it explains how the geometry of ∂Ω affects gradient estimates, which turn out to be sharp.…”

“…In the context of the above, the main goal of this paper is to study the well-posedness and the large time behavior of solutions of capillary-type boundary value problems, i.e., q > 0, of the mean curvature flow with a forcing term and a transport term for the graph case, and to study Neumann boundary problems, q = 1, for the level-set case, on a bounded domain with C 3 boundary, which is not necessarily convex. It generalizes [40] to capillary-type boundary value problems on a nonconvex domain with a force, and generalizes [23] to nonzero Neumann boundary value problems with a transport term.…”