Calculus of Variations and Geometric Measure Theory
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Indecomposable sets of finite perimeter in metric measure spaces

Enrico Pasqualetto (Scuola Normale Superiore)

created by gelli on 26 Jan 2021
modified on 01 Feb 2021

3 feb 2021 -- 17:00   [open in google calendar]

Dipartimento di Matematica (online)

Abstract.

Aim of this talk is to revisit the theory of indecomposable sets of finite perimeter in the setting of isotropic PI spaces. By a PI space we mean a metric measure space that is doubling and supports a weak Poincaré inequality, while isotropicity is a property concerning the Hausdorff-type representation of the perimeter measure. Under these assumptions, we prove a decomposition theorem for sets of finite perimeter into essential connected components and we characterise the extreme points in the space of BV functions. For the decomposition result, we provide two different proofs: via a variational argument, adapted from Ambrosio-Caselles-Masnou-Morel'01, and via Lyapunov Convexity Theorem, inspired by Dolzmann-Müller'95. The result about extreme points generalises a classical theorem by Fleming'60. Based on a joint work with Paolo Bonicatto (University of Warwick) and Tapio Rajala (University of Jyväskylä).

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