Regularity of Sets of Finite Fractional Perimeter and Nonlocal Minimal Surfaces in Metric Measure Spaces

Abstract In the setting of a doubling metric measure space, we study regularity of sets with finite s -perimeter, that is, sets whose characteristic functions have finite Besov energy, with regularity parameter $$0&lt;s&lt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>s</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and exponent $$p=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . Following a result of Visintin in $$\mathbb {R}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> , we provide a sufficient condition for finiteness of the s -perimeter given in terms of the upper Minkowski codimension of the regularized boundary of the set. We also show that if a set has finite s -perimeter, then its measure-theoretic boundary has codimension s Hausdorff measure zero. To the best of our knowledge, this result is new even in the Euclidean setting. By studying certain fat Cantor sets, we provide examples illustrating that the converses of these results do not hold in general. In the doubling metric measure space setting, we then consider minimizers of a nonlocal perimeter functional, extending the definition introduced by Caffarelli, Roquejoffre, and Savin in $$\mathbb {R}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> , and prove existence, uniform density, and porosity results for minimizers.