相关论文: A level-set method for a mean curvature flow with …
We investigate the boundary behavior of variational solutions of Dirichlet problems for prescribed mean curvature equations at smooth boundary points where certain boundary curvature conditions are satisfied (which preclude the existence of…
We introduce a level-set formulation for the mean curvature flow with obstacles and show existence and uniqueness of a viscosity solution. These results generalize a well known viscosity approach for the mean curvature flow without obstacle…
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 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…
In this paper we investigate the numerical approximation of a variant of the mean curvature flow. We consider the evolution of hypersurfaces with normal speed given by $H^k$, $k \ge 1$, where $H$ denotes the mean curvature. We use a level…
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…
In this paper, we propose to use the eikonal equation as a boundary condition when advective or normal flow equations in the level set formulation are solved numerically on polyhedral meshes in the three-dimensional domain. Since the level…
In this paper, we consider the Dirichlet problem for a class of prescribed curvature equations. Both degenerate and non-degenerate cases are considered. The existence of the $C^{1,1}$ regular graphic hypersurfaces with prescribing a class…
We prove two new estimates for the level set flow of mean convex domains in Riemannian manifolds. Our estimates give control - exponential in time - for the infimum of the mean curvature, and the ratio between the norm of the second…
In this paper, we consider the existence of mean curvature type hypersurfaces with prescribed gradient image. Let $\Omega$ and $\tilde{\Omega}$ be uniformly convex bounded domains in $\mathbb{R}^n$ with smooth boundary. We show that there…
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…
In this work, we propose a new evolving geometric flow (called translating mean curvature flow) for the translating solitons of hypersurfaces in $R^{n+1}$. We study the basic properties, such as positivity preserving property, of the…
In this paper, we consider the existence of constant mean curvature hypersurfaces with prescribed gradient image. Let $\Omega$ and $\tilde{\Omega}$ be uniformly convex bounded domains in $\mathbb{R}^n$ with smooth boundary. We show that…
In this paper, we study the Dirichlet problem for a class of prescribed curvature equations in Minkowski space. We prove the existence of smooth spacelike hypersurfaces with a class of prescribed curvature and general boundary data based on…
In this paper, we apply a blow-up method of Schoen and Yau in \cite{SY81} to study a large class of prescribed mean curvature (PMC) Dirichlet problems in $n(n\geq 2)$-dimensional Riemannian manifolds. In this process we establish curvature…
For every closed set $K \subset \mathbb{R}^n$ and every $m \geq 2$, we construct a mean-convex ancient solution to mean curvature flow of hypersurfaces in $\mathbb{R}^{m+n}$, with respect to a smooth Riemannian metric arbitrarily…
In this paper, we prove the existence of classical solutions of the Dirichlet problem for a class of quasi-linear elliptic equations on unbounded domains like a cone or a U-type domain. This problem comes from the study of mean curvature…
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 consider a Dirichlet problem for the mean curvature operator in the Minkowski spacetime, obtaining a necessary and sufficient condition for the existence of a spacelike solution, with prescribed mean curvature, which is the graph of a…
These are lecture notes for the mini-course \textit{PDE and hypersurfaces with prescribed mean curvature} held in Federal University of S\~ao Carlos at the Workshop on Submanifold Theory and Geometric Analysis, August 05 -- 09, 2019. The…