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

  • Wojciech Górny ORCID logo and José M. Mazón ORCID logo EMAIL logo
Published/Copyright: June 29, 2022

Abstract

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 L 1 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.


Communicated by Juha Kinnunen


Funding source: Austrian Science Fund

Award Identifier / Grant number: FR 4083/3-1/I4354

Award Identifier / Grant number: CZ 01/2021

Funding source: Narodowe Centrum Nauki

Award Identifier / Grant number: 2017/27/N/ST1/02418

Award Identifier / Grant number: PGC2018-094775-B-100

Funding statement: The first author has been partially supported by the DFG-FWF project FR 4083/3-1/I4354, by the OeAD-WTZ project CZ 01/2021, and by the project 2017/27/N/ST1/02418 funded by the National Science Centre, Poland. The second author has been partially supported by the Spanish MCIU and FEDER, project PGC2018-094775-B-100.

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Received: 2021-12-17
Revised: 2022-03-12
Accepted: 2022-03-16
Published Online: 2022-06-29
Published in Print: 2024-01-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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