New cases of equality between p-module and p-capacity

Petru Caraman

Annales Polonici Mathematici (1991)

  • Volume: 55, Issue: 1, page 37-56
  • ISSN: 0066-2216

Abstract

top
Let E₀, E₁ be two subsets of the closure D̅ of a domain D of the Euclidean n-space ℝ n and Γ(E₀,E₁,D) the family of arcs joining E₀ to E₁ in D. We establish new cases of equality M p Γ ( E ₀ , E ₁ , D ) = c a p p ( E ₀ , E ₁ , D ) , where M p Γ ( E ₀ , E ₁ , D ) is the p-module of the arc family Γ(E₀,E₁,D), while c a p p ( E ₀ , E ₁ , D ) is the p-capacity of E₀,E₁ relative to D and p > 1. One of these cases is when p = n, E̅₀ ∩ E̅₁ = ∅, E i = E i ' ∪ E i ' ' ∪ E i ' ' ' ∪ F i , E i ' is inaccessible from D by rectifiable arcs, E i ' ' is open relative to D̅ or to the boundary ∂D of D, E i ' ' ' is at most countable, F i is closed (i = 0,1) and D is bounded and m-smooth on (F₀ ∪ F₁) ∩ ∂D.

How to cite

top

Petru Caraman. "New cases of equality between p-module and p-capacity." Annales Polonici Mathematici 55.1 (1991): 37-56. <http://eudml.org/doc/262233>.

@article{PetruCaraman1991,
abstract = {Let E₀, E₁ be two subsets of the closure D̅ of a domain D of the Euclidean n-space $ℝ^n$ and Γ(E₀,E₁,D) the family of arcs joining E₀ to E₁ in D. We establish new cases of equality $M_pΓ(E₀,E₁,D) = cap_p(E₀,E₁,D)$, where $M_pΓ(E₀,E₁,D)$ is the p-module of the arc family Γ(E₀,E₁,D), while $cap_p(E₀,E₁,D)$ is the p-capacity of E₀,E₁ relative to D and p > 1. One of these cases is when p = n, E̅₀ ∩ E̅₁ = ∅, $E_i = E^\{\prime \}_i ∪ E^\{\prime \prime \}_i ∪ E^\{\prime \prime \prime \}_i ∪ F_i$, $E^\{\prime \}_i$ is inaccessible from D by rectifiable arcs, $E^\{\prime \prime \}_i$ is open relative to D̅ or to the boundary ∂D of D, $E^\{\prime \prime \prime \}_i$ is at most countable, $F_i$ is closed (i = 0,1) and D is bounded and m-smooth on (F₀ ∪ F₁) ∩ ∂D.},
author = {Petru Caraman},
journal = {Annales Polonici Mathematici},
keywords = {p-capacity; p-module; -module; -capacity},
language = {eng},
number = {1},
pages = {37-56},
title = {New cases of equality between p-module and p-capacity},
url = {http://eudml.org/doc/262233},
volume = {55},
year = {1991},
}

TY - JOUR
AU - Petru Caraman
TI - New cases of equality between p-module and p-capacity
JO - Annales Polonici Mathematici
PY - 1991
VL - 55
IS - 1
SP - 37
EP - 56
AB - Let E₀, E₁ be two subsets of the closure D̅ of a domain D of the Euclidean n-space $ℝ^n$ and Γ(E₀,E₁,D) the family of arcs joining E₀ to E₁ in D. We establish new cases of equality $M_pΓ(E₀,E₁,D) = cap_p(E₀,E₁,D)$, where $M_pΓ(E₀,E₁,D)$ is the p-module of the arc family Γ(E₀,E₁,D), while $cap_p(E₀,E₁,D)$ is the p-capacity of E₀,E₁ relative to D and p > 1. One of these cases is when p = n, E̅₀ ∩ E̅₁ = ∅, $E_i = E^{\prime }_i ∪ E^{\prime \prime }_i ∪ E^{\prime \prime \prime }_i ∪ F_i$, $E^{\prime }_i$ is inaccessible from D by rectifiable arcs, $E^{\prime \prime }_i$ is open relative to D̅ or to the boundary ∂D of D, $E^{\prime \prime \prime }_i$ is at most countable, $F_i$ is closed (i = 0,1) and D is bounded and m-smooth on (F₀ ∪ F₁) ∩ ∂D.
LA - eng
KW - p-capacity; p-module; -module; -capacity
UR - http://eudml.org/doc/262233
ER -

References

top
  1. [1] P. P. Caraman, p-Capacity and p-modulus, Symposia Math. 18 (1976), 455-484. 
  2. [2] P. P. Caraman, About equality between the p-module and the p-capacity in ℝ n , in: Analytic Functions, Proc. Conf. Błażejewko 1982, Lecture Notes in Math. 1039, Springer, Berlin 1983, 32-83. 
  3. [3] P. P. Caraman, p-module and p-capacity of a topological cylinder, Rev. Roumaine Math. Pures Appl. (1991) (in print). 
  4. [4] P. P. Caraman, p-module and p-capacity in ℝ n , Rev. Roumaine Math. Pures Appl. (in print). 
  5. [5] B. Fuglede, Extremal length and functional completion, Acta Math. 98 (1957), 171-219. Zbl0079.27703
  6. [6] J. Hesse, Modulus and capacity, Ph.D. Thesis, Univ. of Michigan, Ann Arbor, Michigan, 1972. 
  7. [7] J. Hesse, A p-extremal length and p-capacity equality, Ark. Mat. 13 (1975), 131-144. Zbl0302.31009
  8. [8] J. Väisälä, On quasiconformal mappings in space, Ann. Acad. Sci. Fenn. Ser. AI Math. 298 (1961), 1-36. Zbl0096.27506
  9. [9] W. Ziemer, Extremal length and conformal capacity, Trans. Amer. Math. Soc. 126 (1967), 460-473. Zbl0177.34002
  10. [10] W. Ziemer, Extremal length and p-capacity, Michigan Math. J. 16 (1969), 43-51. 

NotesEmbed ?

top

You must be logged in to post comments.