Uniform estimates and limiting arguments for nonlocal minimal surfaces

We consider nonlocal minimal surfaces obtained by a fractional type energy functional, parameterized by s∈(0,1) . We show that the s-energy approaches the perimeter as s → 1−. We also provide density properties and clean ball conditions, which are uniform as s → 1−, and optimal lower bounds obtained by a rearrangement result. Then, we show that s-minimal sets approach sets of minimal perimeter as s → 1−.