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

Petru Caraman¹

¹

Annales Polonici Mathematici, Tome 55 (1991) no. 1, pp. 37-56.

Résumé

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 = ∅, $E_i = E'_i ∪ E''_i ∪ E'''_i ∪ F_i$, $E'_i$ 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.

Détail

DOI : 10.4064/ap-55-1-37-56

Keywords: p-capacity, p-module