L 2 normal velocity implies strong solution for graphical Brakke flows

We prove that if a one-parameter family of varifolds has an L 2 {L²} 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 {∇ 0.569055ptf} , then f has weak derivatives ∂ t ⁡ f , ∇ 2 ⁡ f ∈ L 2 {∂ₜf,∇²f∈ L²} , 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 {Lp,q} and C 0 , α {C0,α} are strong and classical solutions to the forced mean curvature flow equation, respectively.