scholarly journals Semiregular and strongly irregular boundary points for p-harmonic functions on unbounded sets in metric spaces

2021 ◽  
Author(s):  
Anders Björn ◽  
Daniel Hansevi
Keyword(s):  
Harmonic Functions ◽  
Metric Spaces ◽  
Poincaré Inequality ◽  
Doubling Measure ◽  
Poincare Inequality ◽  
Irregular Boundary ◽  
Complete Metric Spaces ◽  
Boundary Points ◽  
Local Properties ◽  
Harmonic Measures

AbstractThe trichotomy between regular, semiregular, and strongly irregular boundary points for $$p$$ p -harmonic functions is obtained for unbounded open sets in complete metric spaces with a doubling measure supporting a $$p$$ p -Poincaré inequality, $$1<p<\infty $$ 1 < p < ∞ . We show that these are local properties. We also deduce several characterizations of semiregular points and strongly irregular points. In particular, semiregular points are characterized by means of capacity, $$p$$ p -harmonic measures, removability, and semibarriers.

scholarly journals Duality of Moduli and Quasiconformal Mappings in Metric Spaces

2020 ◽  
Vol 8 (1) ◽  
pp. 166-181
Author(s):  
Rebekah Jones ◽  
Panu Lahti
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Poincaré Inequality ◽  
Quasiconformal Mappings ◽  
Duality Relation ◽  
Doubling Measure ◽  
Poincare Inequality ◽  
Family Of Curves ◽  
The Family

AbstractWe prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincaré inequality. Then we apply this to show that quasiconformal mappings can be characterized by the fact that they quasi-preserve the modulus of certain families of surfaces.

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 Boundary Regularity for p-Harmonic Functions and Solutions of Obstacle Problems on Unbounded Sets in Metric Spaces

2019 ◽  
Vol 7 (1) ◽  
pp. 179-196
Author(s):  
Anders Björn ◽  
Daniel Hansevi
Keyword(s):  
Harmonic Functions ◽  
Obstacle Problem ◽  
Metric Spaces ◽  
Local Property ◽  
Boundary Regularity ◽  
Obstacle Problems ◽  
Regular Boundary ◽  
Complete Metric Spaces ◽  
Regularity Results

Abstract The theory of boundary regularity for p-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.

scholarly journals Equivalence of two BV classes of functions in metric spaces, and existence of a Semmes family of curves under a 1-Poincaré inequality

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Estibalitz Durand-Cartagena ◽  
Sylvester Eriksson-Bique ◽  
Riikka Korte ◽  
Nageswari Shanmugalingam
Keyword(s):  
Bounded Variation ◽  
Metric Space ◽  
Metric Spaces ◽  
Poincaré Inequality ◽  
The Other ◽  
Functions Of Bounded Variation ◽  
Metric Measure Spaces ◽  
Poincare Inequality ◽  
Family Of Curves ◽  
Measure Spaces

Abstract We consider two notions of functions of bounded variation in complete metric measure spaces, one due to Martio and the other due to Miranda Jr. We show that these two notions coincide if the measure is doubling and supports a 1-Poincaré inequality. In doing so, we also prove that if the measure is doubling and supports a 1-Poincaré inequality, then the metric space supports a Semmes family of curves structure.

Removable sets for the poincare inequality on metric spaces

2000 ◽  
Vol 49 (1) ◽  
pp. 0-0 ◽  
Author(s):  
P. Koskela ◽  
N. Shanmugalingam ◽  
H. Tuominen
Keyword(s):  
Metric Spaces ◽  
Poincaré Inequality ◽  
Poincare Inequality ◽  
Removable Sets

Existence and uniqueness of $$\infty $$-harmonic functions under assumption of $$\infty $$-Poincaré inequality

2018 ◽  
Vol 374 (1-2) ◽  
pp. 881-906 ◽  
Author(s):  
Estibalitz Durand-Cartagena ◽  
Jesús A. Jaramillo ◽  
Nageswari Shanmugalingam
Keyword(s):  
Existence And Uniqueness ◽  
Harmonic Functions ◽  
Poincaré Inequality ◽  
Poincare Inequality

ON THE QUANTIFICATION OF UNIFORM PROPERTIES.PART 2

2001 ◽  
Vol 37 (1-2) ◽  
pp. 169-184
Author(s):  
B. Windels
Keyword(s):  
Metric Spaces ◽  
Uniform Spaces ◽  
Complete Metric Spaces ◽  
The Subject

In 1930 Kuratowski introduced the measure of non-compactness for complete metric spaces in order to measure the discrepancy a set may have from being compact.Since then several variants and generalizations concerning quanti .cation of topological and uniform properties have been studied.The introduction of approach uniform spaces,establishes a unifying setting which allows for a canonical quanti .cation of uniform concepts,such as completeness,which is the subject of this article.

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