Rectifiability of the reduced boundary for sets of finite perimeter over $\operatorname{RCD}(K,N)$ spaces

Elia Bruè; Enrico Pasqualetto; Daniele Semola

doi:10.4171/jems/1217

JournalsjemsVol. 25, No. 2pp. 413–465

Rectifiability of the reduced boundary for sets of finite perimeter over $RCD (K, N)$ spaces

Elia Bruè
Scuola Normale Superiore, Pisa, Italy
Enrico Pasqualetto
University of Jyväskylä, Finland
Daniele Semola
Scuola Normale Superiore, Pisa, Italy

$Rectifiability of the reduced boundary for sets of finite perimeter over $\operatorname{RCD}(K,N)$ spaces cover$

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Abstract

This paper is devoted to the study of sets of finite perimeter in $RCD (K, N)$ metric measure spaces. Its aim is to complete the picture of the generalization of De Giorgi’s theorem within this framework. Starting from the results of Ambrosio et al. (2019) we obtain uniqueness of tangents and rectifiability for the reduced boundary of sets of finite perimeter. As an intermediate tool, of independent interest, we develop a Gauss–Green integration-by-parts formula tailored to this setting. These results are new and non-trivial even in the setting of Ricci limits.

Cite this article

Elia Bruè, Enrico Pasqualetto, Daniele Semola, Rectifiability of the reduced boundary for sets of finite perimeter over $RCD (K, N)$ spaces. J. Eur. Math. Soc. 25 (2023), no. 2, pp. 413–465

DOI 10.4171/JEMS/1217