Isoperimetric inequalities and regularity at shrinking points for parabolic problems

A noncylindrical in time domain QT given by the equation, QT = {(x,t), t∈(t0,T), x∈Ω(t)⊂RN}, is considered. It is assumed that for any τ, t0<τ<T, the domain Qτ is homeomorphic to a cylinder Ω×(t0, τ) where Ω has the classical regularity in parabolic problems. It is also assumed that QT `shrinks' at P, that is Q̄Tintersection{t = T} = P, where P is the point (x = 0, t = T); P is called the `shrinking point' of QT.