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

We study the Neumann and Dirichlet problems for the total variation flow in doubling metric measure spaces supporting a weak Poincaré inequality. We prove existence and uniqueness of weak solutions and study their asymptotic behavior. Furthermore, in the Neumann problem we provide a notion of soluti...

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Bibliographic Details
Published inAdvances in calculus of variations Vol. 17; no. 1; pp. 131 - 164
Main Authors Górny, Wojciech, Mazón, José M.
Format Journal Article
LanguageEnglish
Published De Gruyter 01.01.2024
Subjects
35K90
35K92
49J52
58J35
entropy solutions
Gauss–Green formula
gradient flows
Metric measure spaces
nonsmooth analysis
total variation flow
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Summary:We study the Neumann and Dirichlet problems for the total variation flow in doubling metric measure spaces supporting a weak Poincaré inequality. We prove existence and uniqueness of weak solutions and study their asymptotic behavior. Furthermore, in the Neumann problem we provide a notion of solutions which is valid for 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.
ISSN:1864-8258
1864-8266
DOI:10.1515/acv-2021-0107