Uniform estimates and limiting arguments for nonlocal minimal surfaces

We consider nonlocal minimal surfaces obtained by a fractional type energy functional, parameterized by s is an element of (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(-).