The Neumann and Dirichlet problems for the total variation flow in metric measure spaces

2021/05/24 by Wojciech Górny, Górny, Wojciech, José M. Mazón +1
Computer Science · Mathematics · #35K90 #35K92 #49J52 #58J35 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2105.11424

openalex publication_date 2021/05/24 · openalex created_date 2022/07/25 · openalex updated_date 2026/08/01

Abstract

We study the Neumann and Dirichlet problems for the total variation flow in metric measure spaces. We prove existence and uniqueness of weak solutions and study their asymptotic behaviour. Furthermore, in the Neumann problem we provide a notion of solutions which is valid for L¹ initial data, as well as prove their existence and uniqueness. Our main tools are the first-order linear differential structure due to Gigli and a version of the Gauss-Green formula.

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