We consider the evolution of hypersurfaces with boundary under inverse mean curvature flow. The boundary condition is of Neumann type, i.e. the evolving hypersurface moves along, but stays perpendicular to, a fixed supporting hypersurface. In this setup, we prove existence and uniqueness of weak solutions. Furthermore, we indicate the existence of a monotone quantity which is the analog of the Hawking mass for closed hypersurfaces.

For a given convex cone we consider hypersurfaces with boundary which are star-shaped with respect to the center of the cone and which meet the cone perpendicular. The evolution of those hypersurfaces inside the cone yields a nonlinear parabolic Neumann problem. We show that one can use the convexity of the cone to prove long time existence of this flow. Finally, we show that the hypersurfaces converge smoothly to a piece of the round sphere.

In this article we consider the Dirichlet problem for hypersurfaces of anisotropic prescribed mean curvature H = H (x, u, N ) depending on x ∈ Ω ⊂ R n , the height u of the hypersurface M = graph u over Ω and the unit normal N to M at (x, u). We give a condition relating H and the mean curvature of ∂Ω that guarantees the existence of smooth solutions even for not necessarily convex domains.

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