2014, Dipierro, Serena, Savin, Ovidiu, Valdinoci, Enrico

We consider a nonlocal free boundary problem built by a fractional Dirichlet norm plus a fractional perimeter. Among other results, we prove a monotonicity formula for the minimizers, glueing lemmata, uniform energy bounds, convergence results, a regularity theory for the planar cones and a trivialization result for the flat case. Several classical free boundary problems are limit cases of the one that we consider in this paper.

2014, Castro, Agnese di, Novaga, Matteo, Ruffini, Berardo, Valdinoci, Enrico

We characterize the volume-constrained minimizers of a nonlocal free energy given by the difference of fractional perimeters. Exploiting the quantitative fractional isoperimetric inequality, we show that balls are the unique minimizers if the volume is sufficiently small, while the existence vs. nonexistence of minimizers for large volumes remains open. We also consider the corresponding isoperimetric problem and prove existence and regularity of minimizers.

2016, Cinti, Eleonora, Sinestrari, Carlos, Valdinoci, Enrico

In this paper we consider the evolution of boundaries of sets by a fractional mean curvature flow. We show that, for any dimension n ≥ 2, there exist embedded hypersurfaces in Rn which develop a singularity without shrinking to a point. Such examples are well known for the classical mean curvature flow for n ≥ 3. Interestingly, when n = 2, our result provides instead a counterexample in the nonlocal framework to the well known Grayson's Theorem [17], which states that any smooth embedded curve in the plane evolving by (classical) MCF shrinks to a point. The essential step in our construction is an estimate which ensures that a suitably small perturbation of a thin strip has positive fractional curvature at every boundary point.