On a time-depending Monge–Ampère type equation

In this paper, we prove a comparison result between a solution u(x,t)u(x,t), x∈Ω⊂R2x∈Ω⊂R2, t∈(0,T)t∈(0,T), of a time depending equation involving the Monge–Ampère operator in the plane and the solution of a conveniently symmetrized parabolic equation. To this aim, we prove a derivation formula for the integral of a smooth function g(x,t)g(x,t) over sublevel sets of uu, {x∈Ω:u(x,t)<ϑ}{x∈Ω:u(x,t)<ϑ}, ϑ∈Rϑ∈R, having the same perimeter in R2R2.