scholarly journals Solutions of the divergence operator on John domains

2006 ◽  
Vol 206 (2) ◽  
pp. 373-401 ◽  
Author(s):  
Gabriel Acosta ◽  
Ricardo G. Durán ◽  
María A. Muschietti
Keyword(s):  
Divergence Operator ◽  
John Domains

scholarly journals Unions of John domains

1999 ◽  
Vol 128 (4) ◽  
pp. 1135-1140 ◽  
Author(s):  
Jussi Väisälä
Keyword(s):  
John Domains

scholarly journals Fractional Hardy–Sobolev type inequalities for half spaces and John domains

2018 ◽  
Vol 146 (8) ◽  
pp. 3393-3402 ◽  
Author(s):  
Bartłomiej Dyda ◽  
Juha Lehrbäck ◽  
Antti V. Vähäkangas
Keyword(s):  
Sobolev Type ◽  
John Domains

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.

The Poincaré Inequality and Reverse Doubling Weights

2004 ◽  
Vol 47 (2) ◽  
pp. 206-214 ◽  
Author(s):  
Ritva Hurri-Syrjänen
Keyword(s):  
Large Class ◽  
Poincaré Inequality ◽  
Irregular Domains ◽  
Poincare Inequality ◽  
Poincaré Inequalities ◽  
Doubling Weights ◽  
Poincare Inequalities ◽  
John Domains

AbstractWe show that Poincaré inequalities with reverse doubling weights hold in a large class of irregular domains whenever the weights satisfy certain conditions. Examples of these domains are John domains.

scholarly journals Sharp Capacity Estimates in S-John Domains

2015 ◽  
Vol 43 (2) ◽  
pp. 277-288
Author(s):  
Chang-Yu Guo
Keyword(s):  
John Domains ◽  
Capacity Estimates
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