scholarly journals Equivalent definitions of BV space and of total variation on metric measure spaces

2014 ◽  
Vol 266 (7) ◽  
pp. 4150-4188 ◽  
Author(s):  
Luigi Ambrosio ◽  
Simone Di Marino
Keyword(s):  
Total Variation ◽  
Metric Measure Spaces ◽  
Measure Spaces

scholarly journals Existence of parabolic minimizers to the total variation flow on metric measure spaces

2022 ◽  
Author(s):  
Vito Buffa ◽  
Michael Collins ◽  
Cintia Pacchiano Camacho
Keyword(s):  
Total Variation ◽  
Metric Measure Spaces ◽  
Doubling Condition ◽  
Parabolic Boundary ◽  
Open Set ◽  
Variational Solutions ◽  
Total Variation Flow ◽  
Parabolic Function ◽  
Measure Spaces ◽  
Abstract Notion

AbstractWe give an existence proof for variational solutions u associated to the total variation flow. Here, the functions being considered are defined on a metric measure space $$({\mathcal {X}}, d, \mu )$$ ( X , d , μ ) satisfying a doubling condition and supporting a Poincaré inequality. For such parabolic minimizers that coincide with a time-independent Cauchy–Dirichlet datum $$u_0$$ u 0 on the parabolic boundary of a space-time-cylinder $$\Omega \times (0, T)$$ Ω × ( 0 , T ) with $$\Omega \subset {\mathcal {X}}$$ Ω ⊂ X an open set and $$T > 0$$ T > 0 , we prove existence in the weak parabolic function space $$L^1_w(0, T; \mathrm {BV}(\Omega ))$$ L w 1 ( 0 , T ; BV ( Ω ) ) . In this paper, we generalize results from a previous work by Bögelein, Duzaar and Marcellini by introducing a more abstract notion for $$\mathrm {BV}$$ BV -valued parabolic function spaces. We argue completely on a variational level.

scholarly journals Lower estimates of heat kernels for non-local Dirichlet forms on metric measure spaces

2017 ◽  
Vol 272 (8) ◽  
pp. 3311-3346 ◽  
Author(s):  
Alexander Grigor'yan ◽  
Eryan Hu ◽  
Jiaxin Hu
Keyword(s):  
Dirichlet Forms ◽  
Heat Kernels ◽  
Metric Measure Spaces ◽  
Measure Spaces ◽  
Non Local

scholarly journals A note on the extension of BV functions in metric measure spaces

2008 ◽  
Vol 340 (1) ◽  
pp. 197-208 ◽  
Author(s):  
Annalisa Baldi ◽  
Francescopaolo Montefalcone
Keyword(s):  
Metric Measure Spaces ◽  
Bv Functions ◽  
Measure Spaces

scholarly journals Discretization of integrals on compact metric measure spaces

2021 ◽  
Vol 381 ◽  
pp. 107602
Author(s):  
Martin D. Buhmann ◽  
Feng Dai ◽  
Yeli Niu
Keyword(s):  
Metric Measure Spaces ◽  
Measure Spaces
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