In this note, our aim is to show that families of smooth hypersurfaces of Rⁿ⁺¹ which are all “C¹ –close” enough to a fixed compact, embedded one, have uniformly bounded constants in some relevant inequalities for mathematical analysis, like Sobolev, Gagliardo– Nirenberg and “geometric” Calderòn–Zygmund inequalities.

In this survey we present the state of the art about the asymptotic behavior and stability of the modified Mullins–Sekerka flow and the surface diffusion flow of smooth sets, mainly due to E. Acerbi, N. Fusco, V. Julin and M. Morini. First we discuss in detail the properties of the nonlocal Area functional under a volume constraint, of which the two flows are the gradient flow with respect to suitable norms, in particular, we define the strict stability property for a critical set of such functional and we show that it is a necessary and sufficient condition for minimality under W^2,p–perturbations, holding in any dimension. Then, we show that, in dimensions two and three, for initial sets sufficiently “close” to a smooth strictly stable critical set E, both flows exist for all positive times and asymptotically “converge” to a translate of E.