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

Annales Polonici Mathematici (1991)

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

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

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