We study the evolution in time of smooth sets in the n–dimensional flat torus, such that their boundaries, which are smooth hypersurfaces, move by surface diffusion flow (i.e. the H−1H−1 gradient flow of the Area functional). More precisely, in this talk we present two different proofs of the stability of strictly stable critical sets for the volume–constrained Area functional, under the surface diffusion flow. The first approach is based on suitable energy estimates and compactness arguments, whereas the second one relies on the gradient flow structure of the evolution, in particular, the main tool is the Alexandrov-type inequality combined with the quantitative isoperimetric inequality. Hence, assuming different hypotheses, we prove that if the initial set is sufficiently "close" to a strictly stable critical set, then the flow actually exists globally in time and exponentially converges to a “translation” of the critical set. This is based on joint works with N. Fusco (Univ. Federico II di Napoli & SSM), C. Mantegazza (Univ. Federico II di Napoli & SSM), D. De Gennaro (Univ. Bocconi), A. Kubin (TU Wien) and A. Kubin (Jyväskylä University).
NB: This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006.