Related papers: Inverse mean curvature flow with outer obstacle

We consider the mean curvature flow of compact convex surfaces in Euclidean $3$-space with free boundary lying on an arbitrary convex barrier surface with bounded geometry. When the initial surface is sufficiently convex, depending only on…

In this paper, we discuss uniqueness and backward uniqueness for mean curvature flow of non-compact manifolds. We use an energy argument to prove two uniqueness theorems for mean curvature flow with possibly unbounded curvatures. These…

We introduce a geometric evolution equation of hyperbolic type, which governs the evolution of a hypersurface moving in the direction of its mean curvature vector. The flow stems from a geometrically natural action containing kinetic and…

We prove the existence and uniqueness of strong solutions to the equation $u u_x - u_{yy} = f$ in the vicinity of the linear shear flow, subject to perturbations of the source term and lateral boundary conditions. Since the solutions we…

In this paper we consider a mean curvature flow $V=H+A$ in a high dimensional cylinder $\Omega\times \R$, where, $A$ is a constant, $\Omega$ is a bounded domain in $\R^n$, and, for a hypersurface $y=u(x,t)$ over $\Omega$, $V$ and $H$ denote…

Mean curvature flow evolves isometrically immersed base manifolds $M$ in the direction of their mean curvatures in an ambient manifold $\bar{M}$. If the base manifold $M$ is compact, the short time existence and uniqueness of the mean…

We study the phase field method for the volume preserving mean curvature flow. Given an initial $C^1$ hypersurface we proved the existence of the weak solution for the volume preserving mean curvature flow via the reaction diffusion…

By making use of the nice behavior of Hawking masses of slices of a weak solution of inverse mean curvature flow in three dimensional asymptotically hyperbolic manifolds, we are able to show that each slice of the flow is star-shaped after…

We construct weak solutions of the anisotropic inverse mean curvature flow (A-IMCF) under very mild assumptions both on the anisotropy (which is simply a norm in $\mathbb R^N$ with no ellip\-ticity nor smoothness requirements, in order to…

We consider the inverse mean curvature flow in smooth Riemannian manifolds of the form $([R_{0},\infty)\times S^n,\bar{g})$ with metric $\bar{g}=dr^2+{\vartheta}^2(r){\sigma}$ and non-positive radial sectional curvature. We prove, that for…

In this paper, we study the evolution of submannifold moving by mean curvature minus a external force field. We prove that the flow has a long-time smooth solution for all time under almost optimal conditions. Those conditions are that the…

Thermodynamically consistent models for two-phase flow in porous media have attracted significant attention in recent years. In this paper, we prove the existence, uniqueness and regularity of the weak solution to such a recent model…

In [5], S\'aez and Schn\"urer studied the graphical mean curvature flow of complete hypersurfaces defined on subsets of Euclidean space. They obtained long time existence. Moreover, they provided a new interpretation of weak mean curvature…

By a symmetric double graph we mean a hypersurface which is mirror-symmetric and the two symmetric parts are graphs over the hyperplane of symmetry. We prove that there is a weak solution of mean curvature flow that preserves these…

We provide some counterexamples concerning the uniqueness and regularity of weak solutions to the initial-boundary value problem for gradient flows of certain strongly polyconvex functionals by showing that such a problem can possess a…

We study inverse mean curvature flows of starshaped, mean convex hypersurfaces in warped product manifolds with a positive warping factor $\varphi(r)$. If $\varphi'(r)>0$ and $\varphi''(r)\geq 0$, we show that these flows exist for all…

In this note we consider asymptotically flat manifolds with non-negative scalar curvature and an inner boundary which is an outermost minimal surface. We show that there exists an upper bound on the mean curvature of a constant mean…

In this work we study solutions of the prescribed mean curvature equation over a general domain that do not necessarily attain the given boundary data. To such a solution, we can naturally associate a current with support in the closed…

In this article, we prove the mean convex neighborhood conjecture for the mean curvature flow of surfaces in $\mathbb{R}^3$. Namely, if the flow has a spherical or cylindrical singularity at a space-time point $X=(x,t)$, then there exists a…

In this paper, we study surfaces which evolve by anisotropic mean curvature flow with contact angle boundary condition over a strictly convex domain in $\mathbb{R}^2$. We establish a prior gradient estimate for smooth solutions to this…