scholarly journals A compactness result for BV functions in metric spaces

2019 ◽  
Vol 44 (1) ◽  
pp. 329-339 ◽  
Author(s):  
Sebastiano Don ◽  
Davide Vittone
Keyword(s):  
Metric Spaces ◽  
Compactness Result ◽  
Bv Functions

The variational 1-capacity and BV functions with zero boundary values on doubling metric spaces

2018 ◽  
Vol 0 (0) ◽  
Author(s):  
Panu Lahti
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Poincaré Inequality ◽  
Doubling Measure ◽  
Boundary Values ◽  
Poincare Inequality ◽  
Compactly Supported ◽  
Bv Functions

AbstractIn the setting of a metric space that is equipped with a doubling measure and supports a Poincaré inequality, we define and study a class of{\mathrm{BV}}functions with zero boundary values. In particular, we show that the class is the closure of compactly supported{\mathrm{BV}}functions in the{\mathrm{BV}}norm. Utilizing this theory, we then study the variational 1-capacity and its Lipschitz and{\mathrm{BV}}analogs. We show that each of these is an outer capacity, and that the different capacities are equal for certain sets.

scholarly journals Functions of Bounded Variation on Complete and Connected One-Dimensional Metric Spaces

2020 ◽  
Author(s):  
Panu Lahti ◽  
Xiaodan Zhou
Keyword(s):  
Bounded Variation ◽  
Hausdorff Measure ◽  
Metric Spaces ◽  
General Setting ◽  
Functions Of Bounded Variation ◽  
One Dimensional ◽  
Underlying Space ◽  
Finite Perimeter ◽  
Bv Functions ◽  
Definition Of

Abstract In this paper, we study functions of bounded variation on a complete and connected metric space with finite one-dimensional Hausdorff measure. The definition of BV functions on a compact interval based on pointwise variation is extended to this general setting. We show this definition of BV functions is equivalent to the BV functions introduced by Miranda [18]. Furthermore, we study the necessity of conditions on the underlying space in Federer’s characterization of sets of finite perimeter on metric measure spaces. In particular, our examples show that the doubling and Poincaré inequality conditions are essential in showing that a set has finite perimeter if the codimension one Hausdorff measure of the measure-theoretic boundary is finite.

Crystallography in two-dimensional metric spaces*

1969 ◽  
Vol 130 (1-6) ◽  
pp. 277-303 ◽  
Author(s):  
Aloysio Janner ◽  
Edgar Ascher
Keyword(s):  
Metric Spaces ◽  
Two Dimensional
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