Eleonora Cinti

Università di Bologna

Flatness results for stable solutions to some nonlocal problems

We study stable critical points to two, closely related, nonlocal functionals: the fractional perimeter and the fractional Allen–Cahn energy. In particular, we establish optimal energy estimates, whose local analogue is still unknown. These estimates, together with other ingredients (such as density estimates and a monotonicity formula) allow to reduce the classification of stable critical points for both functionals to the classification of stable hypercones. These results have been obtained in collaboration with X. Cabré and J. Serra.