The logarithmic Sobolev capacity

2021 ◽  
Vol 392 ◽  
pp. 107993
Author(s):  
Liguang Liu ◽  
Suqing Wu ◽  
Jie Xiao ◽  
Wen Yuan
Keyword(s):  
Sobolev Capacity

scholarly journals Dichotomy of global capacity density in metric measure spaces

2018 ◽  
Vol 11 (4) ◽  
pp. 387-404 ◽  
Author(s):  
Hiroaki Aikawa ◽  
Anders Björn ◽  
Jana Björn ◽  
Nageswari Shanmugalingam
Keyword(s):  
Metric Spaces ◽  
Exact Formula ◽  
Doubling Measure ◽  
Metric Measure Spaces ◽  
Geodesic Metric ◽  
Measure Spaces ◽  
Variational Capacity ◽  
John Domains ◽  
Superlevel Sets ◽  
Sobolev Capacity

AbstractThe variational capacity {\operatorname{cap}_{p}} in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every {E\subset{\mathbb{R}}^{n}},\inf_{x\in{\mathbb{R}}^{n}}\frac{\operatorname{cap}_{p}(E\cap B(x,r),B(x,2r))}% {\operatorname{cap}_{p}(B(x,r),B(x,2r))}is either zero or tends to 1 as {r\to\infty}. We prove that this property still holds in unbounded complete geodesic metric spaces equipped with a doubling measure supporting a p-Poincaré inequality, but that it can fail in nongeodesic metric spaces and also for the Sobolev capacity in {{\mathbb{R}}^{n}}. It turns out that the shape of balls impacts the validity of the density dichotomy. Even in more general metric spaces, we construct families of sets, such as John domains, for which the density dichotomy holds. Our arguments include an exact formula for the variational capacity of superlevel sets for capacitary potentials and a quantitative approximation from inside of the variational capacity.

scholarly journals The variable exponent BV-Sobolev capacity

2012 ◽  
Vol 27 (1) ◽  
pp. 13-40 ◽  
Author(s):  
Heikki Hakkarainen ◽  
Matti Nuortio
Keyword(s):  
Variable Exponent ◽  
Sobolev Capacity

A direct approach to Orlicz–Sobolev capacity

2005 ◽  
Vol 60 (1) ◽  
pp. 129-147 ◽  
Author(s):  
Matthew Rudd
Keyword(s):  
Direct Approach ◽  
Sobolev Capacity

scholarly journals Sobolev capacity on the spaceW1, p(⋅)(ℝn)

2003 ◽  
Vol 1 (1) ◽  
pp. 17-33 ◽  
Author(s):  
Petteri Harjulehto ◽  
Peter Hästö ◽  
Mika Koskenoja ◽  
Susanna Varonen
Keyword(s):  
Hausdorff Dimension ◽  
Sobolev Space ◽  
Variable Exponent ◽  
Choquet Capacity ◽  
Sobolev Capacity

We define Sobolev capacity on the generalized Sobolev spaceW1, p(⋅)(ℝn). It is a Choquet capacity provided that the variable exponentp:ℝn→[1,∞)is bounded away from 1 and∞. We discuss the relation between the Hausdorff dimension and the Sobolev capacity. As another application we study quasicontinuous representatives in the spaceW1, p(⋅)(ℝn).

scholarly journals Comparisons of relative BV-capacities and Sobolev capacity in metric spaces

2011 ◽  
Vol 74 (16) ◽  
pp. 5525-5543 ◽  
Author(s):  
Heikki Hakkarainen ◽  
Nageswari Shanmugalingam
Keyword(s):  
Metric Spaces ◽  
Sobolev Capacity

scholarly journals Anisotropic Sobolev Capacity with Fractional Order

2017 ◽  
Vol 69 (4) ◽  
pp. 873-889 ◽  
Author(s):  
Jie Xiao ◽  
Deping Ye
Keyword(s):  
Fractional Order ◽  
Sobolev Spaces ◽  
Lebesgue Space ◽  
Sobolev Class ◽  
Minkowski Inequality ◽  
Fractional Sobolev Spaces ◽  
Geometric Quantity ◽  
Asymptotically Optimal ◽  
Order Estimation ◽  
Sobolev Capacity

AbstractIn this paper, we introduce the anisotropic Sobolev capacity with fractional order and develop some basic properties for this new object. Applications to the theory of anisotropic fractional Sobolev spaces are provided. In particular, we give geometric characterizations for a nonnegative Radon measure μ that naturally induces an embedding of the anisotropic fractional Sobolev class into the μ-based-Lebesgue-space with 0 < β ≤ n. Also, we investigate the anisotropic fractional α-perimeter. Such a geometric quantity can be used to approximate the anisotropic Sobolev capacity with fractional order. Estimation on the constant in the related Minkowski inequality, which is asymptotically optimal as α →0+, will be provided.

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