Mean curvature motion of graphs with constant contact angle at a free boundary

Mean Curvature Motion of Graphs with Constant Contact Angle at a Free Boundary

We consider the motion by mean curvature of an n-dimensional graph over a time-dependent domain in Rn, intersecting Rn at a constant angle. In the general case, we prove local existence for the corresponding quasilinear parabolic equation with a free boundary, and derive a continuation criterion based on the second fundamental form. If the initial graph is concave, we show this is preserved, an...

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

Constant mean curvature surfaces with boundary on a sphere

In this article we study the shape of a compact surface of constant mean curvature of Euclidean space whose boundary is contained in a round sphere. We consider the case that the boundary is prescribed or that the surface meets the sphere with a constant angle. We study under what geometric conditions the surface must be spherical. Our results apply in many scenarios in physics where in absence...

A note on radial graphs with constant mean curvature

Let be a smooth domain on the unit sphere S whose closure is contained in an open hemisphere and denote by H the mean curvature of ∂ as a submanifold of with respect to the inward unit normal. It is proved that for each real number H that satisfies inf H > −H ≥ 0, there exists a unique radial graph on bounded by ∂ with constant mean curvature H . The orientation on the graph is based on the nor...

Mean Curvature Motion of Non-parametric Hypersurfaces with Contact Angle Condition

describes the evolution of graph(u(·, t)) by its mean curvature in the direction of the unit normal with prescribed contact angle (given by cos−1 φ) at boundary. This problem has been studied by G. Huisken [3] for φ ≡ 0, that is, the surfaces have vertical contact angle at the boundary, and by Altschuler and Wu [1] for the case that n = 2 and Ω is strictly convex. The main result of [3] states ...