| Summary: | In this survey we present the state of the art about the asymptotic behavior and stability of the <i>modified Mullins</i>–<i>Sekerka flow</i> and the <i>surface diffusion flow</i> 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 <i>strict stability</i> 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 <i>strictly stable critical</i> set $ E $, both flows exist for all positive times and asymptotically "converge" to a translate of $ E $.
|
|---|