The Total Variation Flow in Metric Random Walk Spaces
| Autor: | José M. Mazón, Marcos Solera, Julián Toledo |
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| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
Markov chain Applied Mathematics 010102 general mathematics Space (mathematics) Random walk 01 natural sciences 05C80 35R02 05C21 45C99 26A45 Sobolev inequality 010101 applied mathematics Mathematics - Analysis of PDEs Flow (mathematics) Metric (mathematics) FOS: Mathematics Uniqueness 0101 mathematics Isoperimetric inequality Analysis Mathematics Analysis of PDEs (math.AP) |
| DOI: | 10.48550/arxiv.1905.01130 |
| Popis: | In this paper we study the Total Variation Flow (TVF) in metric random walk spaces, which unifies into a broad framework the TVF on locally finite weighted connected graphs, the TVF determined by finite Markov chains and some nonlocal evolution problems. Once the existence and uniqueness of solutions of the TVF has been proved, we study the asymptotic behaviour of those solutions and, with that aim in view, we establish some inequalities of Poincar\'{e} type. In particular, for finite weighted connected graphs, we show that the solutions reach the average of the initial data in finite time. Furthermore, we introduce the concepts of perimeter and mean curvature for subsets of a metric random walk space and we study the relation between isoperimetric inequalities and Sobolev inequalities. Moreover, we introduce the concepts of Cheeger and calibrable sets in metric random walk spaces and characterize calibrability by using the $1$-Laplacian operator. Finally, we study the eigenvalue problem whereby we give a method to solve the optimal Cheeger cut problem. |
| Databáze: | OpenAIRE |
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