L ² normal velocity implies strong solution for graphical Brakke flows

Abstract

We prove that if a one-parameter family of varifolds has an

normal velocity v in the sense of Brakke, and if the family is represented as the graph of a continuous function f with continuous spatial derivative

∇ ⁡ f {\nabla\kern 0.569055ptf}

, then f has weak derivatives

∂ t ⁡ f , ∇ 2 ⁡ f ∈ L 2 {\partial_{t}f,\nabla^{2}f\in L^{2}}

, and v coincides with the usual normal velocity of the graph. Moreover, by combining this result with parabolic regularity theory, we show that graphical Brakke flows with forcing term in

L p , q {L^{p,q}}

and

C 0 , α {C^{0,\alpha}}

are strong and classical solutions to the forced mean curvature flow equation, respectively.