相关论文: $C^{1,\alpha}$ theory for the prescribed mean curv…
We study and solve the Dirichlet problem for graphs of prescribed mean curvature in $\mathbb R^{n+1}$ over general domains $\Omega$ without requiring a mean convexity assumption. By using pieces of nodoids as barriers we first give…
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 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…
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…
In this paper, we solve the Dirichlet problem with continuous boundary data for the Lagrangian mean curvature equation on a uniformly convex, bounded domain in $\mathbb{R}^n$.
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 consider prescribed mean curvature equations whose solutions are minimal surfaces, constant mean curvature surfaces, or capillary surfaces. We consider both Dirichlet boundary conditions for Plateau problems and nonlinear Neumann…
In this paper we prove existence of nonnegative bounded solutions for the non-autonomous prescribed mean curvature problem in non-parametric form on an open bounded domain $\Omega$ of $\mathbb{R}^N$. The mean curvature, that depends on the…
In 1976, Leon Simon showed that if a compact subset of the boundary of a domain is smooth and has negative mean curvature, then the non-parametric least area problem with Lipschitz continuous Dirichlet boundary data has a generalized…
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 study the prescribed mean curvature equation for $t$-graphs in a Riemannian Heisenberg group of arbitrary dimension. We characterize the existence of classical solutions in a bounded domain without imposing Dirichlet boundary data, and…
In the homogeneous manifold $\mathbb{E}(-1,\tau),$ for $-\tfrac{1}{2}<H<\tfrac{1}{2},$ {we define a new product compactification in which the slices $\left\{t=c\right\}_{c\in\R}$ are rotational $H$-surfaces. This product compatification is…
We study a prescribed mean curvature problem where we seek a surface whose mean curvature vector coincides with the normal component of a given vector field. We prove that the problem has a solution near a graphical minimal surface if the…
In this paper, we establish the existence of prescribed mean curvature (PMC) hypersurfaces in conformal product manifolds with (possibly empty) $C^{1,\alpha}$ fixed graphical boundaries under a barrier condition. This result generalizes…
We prove that a surface of prescribed mean curvature ($H$-surface) with free boundary on a two-dimensional $C^2$-manifold belongs to $C^{1,\frac{1}{2}}$ up to that the boundary, provided it is a-priori continuous. We allow the $H$-surface…
In this paper, we establish global $C^{1,\alpha}$-regularity for bounded generalized solutions of elliptic equations in divergence form with Musielak-Orlicz growth and subject to Dirichlet or Neumann boundary conditions. In fact, our…
We study $n$-dimensional area-minimizing currents $T$ in $\mathbb{R}^{n+1},$ with boundary $\partial T$ satisfying two properties: $\partial T$ is locally a finite sum of $(n-1)$-dimensional $C^{1,\alpha}$ orientable submanifolds which only…
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…
We establish the existence of hypersurfaces with constant mean curvature and a prescribed boundary in Euclidean space, represented as radial graphs over domains of the unit sphere. Under the assumptions that the mean curvature 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…