Related papers: Two dimensional anisotropic mean curvature flow wi…

We propose a level-set method for a mean curvature flow whose boundary is prescribed by interpreting the boundary as an obstacle. Since the corresponding obstacle problem is globally solvable, our method gives a global-in-time level-set…

We consider a system of three surfaces, graphs over a bounded domain in ${\mathbb R}^2$, intersecting along a time-dependent curve and moving by mean curvature while preserving the pairwise angles at the curve of intersection (equal to…

In this paper we study nonparametric mean curvature type flows in $M\times\mathbb{R}$ which are represented as graphs $(x,u(x,t))$ over a domain in a Riemannian manifold $M$ with prescribed contact angle. The speed of $u$ is the mean…

We show the existence of a smooth solution for the flow deformed by the square root of the scalar curvature multiplied by a positive anisotropic factor $\psi$ given a strictly convex initial hypersurface in Euclidean space suitably pinched.…

We investigate the evolution of strictly convex hypersurfaces driven by the $k$-Hessian curvature flow, subject to the second boundary condition. We first explore the translating solutions corresponding to this boundary value problem. Next,…

We consider the motion by mean curvature of an $n$-dimensional graph over a time-dependent domain in $\mathbb{R}^n$, intersecting $\mathbb{R}^n$ at a constant angle. In the general case, we prove local existence for the corresponding…

We propose an alternative condition for the solvability of the Dirichlet problem for the minimal surface equation that applies to non-mean convex domains. We introduce a structural condition, obtained from a second-order ordinary…

Given a mean curvature flow of compact, embedded $C^2$ surfaces satisfying Neumann free boundary condition on a mean convex, smooth support surface in 3-dimensional Euclidean space, we show that it can be extended as long as its mean…

We construct new examples of immortal mean curvature flow of smooth embedded connected hypersurfaces in closed manifolds, which converge to minimal hypersurfaces with multiplicity $2$ as time approaches infinity.

We consider $n$-dimensional hypersurfaces flowing by mean curvature flow with Neumann free boundary conditions supported on a smooth support surface. We show that the Hausdorff $n$-measure of the singular set is zero. In fact, we consider…

We study the long time existence theory for a non local flow associated to a free boundary problem for a trapped non liquid drop. The drop has free boundary components on two horizontal plates and its free energy is anisotropic and axially…

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 study the Dirichlet problem of translating mean curvature equations over domains in Riemannian manifolds with dimension $n$. Imitating the generalized solution theory of Miranda-Giusti, we define a new conformal area…

We study the Dirichlet problem associated to the equation for self-similar surfaces for graphs over the Euclidean plane with a disk removed. We show the existence of a solution provided the boundary conditions on the boundary circle are…

We investigate the convergence of the mean curvature flow of arbitrary codimension in Riemannian manifolds with bounded geometry. We prove that if the initial submanifold satisfies a pinching condition, then along the mean curvature flow…

We study a variant of the mean curvature flow for closed, convex hypersurfaces where the normal velocity is a nonhomogeneous function of the principal curvatures. We show that if the initial hypersurface satisfies a certain pinching…

In this paper, we prove the short-time existence of hyperbolic inverse (mean) curvature flow (with or without the specified forcing term) under the assumption that the initial compact smooth hypersurface of $\mathbb{R}^{n+1}$…

We develop a new boundary condition for the weak inverse mean curvature flow, which gives canonical and non-trivial solutions in bounded domains. Roughly speaking, the boundary of the domain serves as an outer obstacle, and the evolving…

We consider a fully nonlinear parabolic equation with nonlinear Neumann type boundary condition, and show that the longtime existence and convergence of the flow. Finally we apply this study to the boundary value problem for minimal…

We study the convergence of an axially symmetric hypersurface evolving by volume preserving mean curvature flow. Assuming the surface is not pinching off along the axis at any time during the flow, and without any additional conditions, as…