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
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
Funding source: Ministerio de Ciencia e Innovación
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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Articles in the same Issue
- Frontmatter
- On the Hölder regularity of all extrema in Hilbert’s 19th Problem
- A partially overdetermined problem in domains with partial umbilical boundary in space forms
- On the existence of canonical multi-phase Brakke flows
- Concavity properties for solutions to p-Laplace equations with concave nonlinearities
- On the Almgren minimality of the product of a paired calibrated set and a calibrated manifold of codimension 1
- The Neumann and Dirichlet problems for the total variation flow in metric measure spaces
- No Lavrentiev gap for some double phase integrals
- Characterizations of the viscosity solution of a nonlocal and nonlinear equation induced by the fractional p-Laplace and the fractional p-convexity
- Properties of the free boundaries for the obstacle problem of the porous medium equations
- A generalized fractional Pohozaev identity and applications
Articles in the same Issue
- Frontmatter
- On the Hölder regularity of all extrema in Hilbert’s 19th Problem
- A partially overdetermined problem in domains with partial umbilical boundary in space forms
- On the existence of canonical multi-phase Brakke flows
- Concavity properties for solutions to p-Laplace equations with concave nonlinearities
- On the Almgren minimality of the product of a paired calibrated set and a calibrated manifold of codimension 1
- The Neumann and Dirichlet problems for the total variation flow in metric measure spaces
- No Lavrentiev gap for some double phase integrals
- Characterizations of the viscosity solution of a nonlocal and nonlinear equation induced by the fractional p-Laplace and the fractional p-convexity
- Properties of the free boundaries for the obstacle problem of the porous medium equations
- A generalized fractional Pohozaev identity and applications