Interior gradient estimates for anisotropic mean-curvature flow

Page 1 INTERIOR GRADIENT ESTIMATES FOR MEAN CURVATURE EQUATIONS

In this paper we give a simple proof for the interior gradient estimate for curvature and Hessian equations. We also derive a Liouville type result for these equations. §0. Introduction The interior gradient estimate for the prescribed mean curvature equation has been extensively studied, see [9] and the references therein. For high order mean curvature equations it has also been obtained in [1...

Global boundedness, interior gradient estimates, and boundary regularity for the mean curvature equation with boundary conditions

where ν is the outward pointing unit normal of ∂Ω, and where cosθ is a given function on ∂Ω. (Thus, in the capillarity problem, we are considering geometrically a function u in Ω̄ whose graph has the prescribed mean curvature H and which meets the boundary cylinder in the prescribed angle θ.) Here, H = H(x,t) is assumed to be a given locally Lipschitz function in Ω×R satisfying the structural co...

Anisotropic and Crystalline Mean Curvature Flow

Interior Estimates and Longtime Solutions for Mean Curvature Flow of Noncompact Spacelike Hypersurfaces in Minkowski Space

Spacelike hypersurfaces with prescribed mean curvature have played a major role in the study of Lorentzian manifolds Maximal mean curvature zero hypersurfaces were used in the rst proof of the positive mass theorem Constant mean curvature hypersurfaces provide convenient time gauges for the Einstein equations For a survey of results we refer to In and it was shown that entire solutions of the m...

Sharp Estimates for Mean Curvature Flow of Graphs

A one-parameter family of smooth hypersurfaces {Mt} ⊂ R flows by mean curvature if zt = H(z) = ∆Mtz , (0.1) where z are coordinates on R and H = −Hn is the mean curvature vector. In this note, we prove sharp gradient and area estimates for graphs flowing by mean curvature. Thus, each Mt is assumed to be the graph of a function u(·, t). So, if z = (x, y) with x ∈ R, then Mt is given by y = u(x, ...