On surfaces of prescribed weighted mean curvature

Utilising a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and von der Mosel. Next we study graphs of prescribed weighted mean curvature, for which a quasilinear elliptic equation is proved. Using this equation, we can show height and boundary gradient estimates. Finally, we solve the Dirichlet problem for graphs of prescribed weighted mean curvature.     

Boundary regularity for minimizing currents with prescribed mean curvature

We prove complete boundary regularity for energy minimizing integer multiplicity rectifiablen currents in ⁿ⁺¹ of prescribed mean curvatureH with boundaryB= represented by an oriented smooth submanifold of dimensionn – 1 in sun+1. We also give applications to the Plateau problem for surfaces with prescribed mean curvature.This article was processed by the author using the LaT_EX style filepljour1 from Springer-Verlag.     

The prescribed mean curvature equation for nonparametric surfaces

We study the prescribed mean curvature equation for nonparametric surfaces, obtaining existence and uniqueness results in the Sobolev space W^2,p. We also prove that under appropriate conditions the set of surfaces of mean curvature H is a connected subset of W^2,p. Moreover, we obtain existence results for a boundary value problem which generalizes the one-dimensional periodic problem for the mean curvature equation.     

Existence of prescribed mean curvature graphs in hyperbolic space

In this paper we are concerned with questions of existence and uniqueness of graph-like prescribed mean curvature hypersurfaces in hyperbolic space ?ⁿ⁺¹. In the half-space setting, we will study radial graphs over the totally geodesic hypersurface . We prove the following existence result: Let be a bounded domain of class and let . If everywhere on , where denotes the hyperbolic mean curvature of the cylinder over , then for every there is a unique graph over with mean curvature attaining the boundary values on . Further we show the existence of smooth boundary data such that no solution exists in case of for some under the assumption that has a sign.     

Conformal metrics with prescribed boundary mean curvature on balls

We provide a variety of classes of functions that can be realized as the mean curvature on the boundary of the standard n dimensional ball, n ≥ 3, with respect to some scalar flat metric. Because of the presence of some critical nonlinearity, blow up phenomena occur and existence results are highly nontrivial since one has to overcome topological obstructions. Our approach consists of, on one hand, developing a Morse theoretical approach to this problem through a Morse-type reduction of the associated Euler–Lagrange functional in a neighborhood of its critical points at Infinity and, on the other hand, extending to this problem some topological invariants introduced by A. Bahri in his study of Yamabe type problems on closed manifolds.     

Finite element approximations to surfaces of prescribed variable mean curvature

We give an algorithm for finding finite element approximations to surfaces of prescribed variable mean curvature, which span a given boundary curve. We work in the parametric setting and prove optimal estimates in the H¹ norm. The estimates are verified computationally.     

On the dirichlet problem for surfaces of prescribed mean curvature

A necessary and sufficient condition is given for the solvability of the Dirichlet problem for surfaces of prescribed mean curvature, and global regularity of the solution is studied.     

Estimate for evolutionary surfaces of prescribed mean curvature and the convergence

In the paper, we will discuss the gradient estimate for the evolutionary surfaces of prescribed mean curvature with Neumann boundary value under the condition $f_\tau\ge -\kappa$, which is the same as the one in the interior estimate by K. Ecker and generalizes the condition $f_\tau\ge 0$ studied by Gerhardt etc. Also, based on the elliptic result obtained recently, we will show the longtime behavior of surfaces moving by the velocity being equal to the mean curvature.