scholarly journals Existence of variational solutions to a Cauchy–Dirichlet problem with time-dependent boundary data on metric measure spaces

2020 ◽  
Author(s):  
Michael Collins
Keyword(s):  
Dirichlet Problem ◽  
Boundary Data ◽  
Time Dependent ◽  
Metric Measure Spaces ◽  
Variational Solutions ◽  
Measure Spaces

Dirichlet problem at infinity on Gromov hyperbolic metric measure spaces

2007 ◽  
Vol 339 (1) ◽  
pp. 101-134 ◽  
Author(s):  
Ilkka Holopainen ◽  
Urs Lang ◽  
Aleksi Vähäkangas
Keyword(s):  
Dirichlet Problem ◽  
Hyperbolic Metric ◽  
Metric Measure Spaces ◽  
Gromov Hyperbolic ◽  
Measure Spaces ◽  
Dirichlet Problem At Infinity

scholarly journals Notions of Dirichlet problem for functions of least gradient in metric measure spaces

2019 ◽  
Vol 35 (6) ◽  
pp. 1603-1648 ◽  
Author(s):  
Riikka Korte ◽  
Panu Lahti ◽  
Xining Li ◽  
Nageswari Shanmugalingam
Keyword(s):  
Dirichlet Problem ◽  
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.

Global higher integrability for non-quadratic parabolic quasi-minimizers on metric measure spaces

2017 ◽  
Vol 10 (3) ◽  
pp. 267-301 ◽  
Author(s):  
Yohei Fujishima ◽  
Jens Habermann
Keyword(s):  
Boundary Data ◽  
Measure Space ◽  
Metric Measure Spaces ◽  
Optimal Conditions ◽  
Open Domain ◽  
Higher Integrability ◽  
Global Higher Integrability ◽  
Starting Point ◽  
Measure Spaces ◽  
Stability Issues

AbstractWe prove up-to-the-boundary higher integrability estimates for parabolic quasi-minimizers on a domain \Omega_{T}= Ω \times (0,T), where Ω denotes an open domain in a doubling metric measure space which supports a Poincaré inequality. The higher integrability for upper gradients is shown globally and under optimal conditions on the boundary \partialΩ of the domain as well as on the boundary data itself. This is a starting point for a further discussion on parabolic quasi-minima on metric measure spaces, such as for example regularity or stability issues.

scholarly journals Domains in Metric Measure Spaces with Boundary of Positive Mean Curvature, and the Dirichlet Problem for Functions of Least Gradient

2018 ◽  
Vol 29 (4) ◽  
pp. 3176-3220
Author(s):  
Panu Lahti ◽  
Lukáš Malý ◽  
Nageswari Shanmugalingam ◽  
Gareth Speight
Keyword(s):  
Dirichlet Problem ◽  
Mean Curvature ◽  
Metric Measure Spaces ◽  
Measure Spaces

Heat Flow on Time-Dependent Metric Measure Spaces and Super-Ricci Flows

2018 ◽  
Vol 71 (12) ◽  
pp. 2500-2608 ◽  
Author(s):  
Eva Kopfer ◽  
Karl-Theodor Sturm
Keyword(s):  
Heat Flow ◽  
Time Dependent ◽  
Metric Measure Spaces ◽  
Ricci Flows ◽  
Measure Spaces

scholarly journals Stability and Continuity of Functions of Least Gradient

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
H. Hakkarainen ◽  
R. Korte ◽  
P. Lahti ◽  
N. Shanmugalingam
Keyword(s):  
Dirichlet Problem ◽  
Maximum Principle ◽  
Measure Zero ◽  
Metric Measure Spaces ◽  
Local Minimizers ◽  
Stability Properties ◽  
Area Functional ◽  
Measure Spaces

Abstract In this note we prove that on metric measure spaces, functions of least gradient, as well as local minimizers of the area functional (after modification on a set of measure zero) are continuous everywhere outside their jump sets. As a tool, we develop some stability properties of sequences of least gradient functions. We also apply these tools to prove a maximum principle for functions of least gradient that arise as solutions to a Dirichlet problem.

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