Estimation of the second derivatives for surfaces evolving under the action of their principal curvatures

where ∂′′Q = ∂Ω× [0, T ], Ω(0) = {z = (x, t) : x ∈ Ω, t = 0}, and Ω is a domain in R with a smooth boundary, which only for the sake of simplicity we assumed to be bounded. In (1), (2), g and φ are smooth known functions of z, defined on Q, and k(u) = (k1(u), . . . , kn(u)), where ki(u)(z) are the principal curvatures of the graph Tt: x0 = u(x, t), x ∈ Ω, of the sought function u( · , t) : Ω → R for fixed t ∈ [0, T ].