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

ArticleOriginal scientific text

Title

Authors ¹

Institute of Mathematics, Romanian Academy of Sciences, Iaşi Branch, Bdul Copou 8, Iaşi, Romania

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) = \cap_{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

\cap_{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} \cup E''_{i} \cup E'''_{i} \cup 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.

p-capacity, p-module

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