ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: not given Caraman, Petru, Institute “Octav Mayer”, Romanian Academy, Iaşi, Romania Relations between p-module and p-capacity in R^{n} Let \overline{R^{n}} be one-point compactification of Euclidean n-space R^{n}, E arbitrary sets, F closed sets, Ē the closure of E. The p-module of an arc family Γ is

M_{p}Γ = inf_{ρ∈F (Γ)} ∫ ρ^{p}dm (p > 1), ,

where F (Γ) = {ρ : R^{n} → [0, ∞]; Borel measurable and ∫_{γ} pdH^{1} ≥ 1 ∀γ ∈ Γ} and m and H^{1} are n-dimensional Lebesgue and linear Hausdorff measures, respectively. MΓ = M_{n}Γ is called the module of Γ. The p-capacity of E_{0}, E_{1} ⊂ D̄ ⊂ \overline{R^{n}} relatively to D is

cap_{p}(E_{0}, E_{1}, D) = inf_{u∈U} ∫_{D} |∇u|^{p}dm (p > 1) ,

where U = {u : D ∪ E_{0} ∪ E_{1} → [0, 1]; u continuous, u_{|_{D−{∞}}} locally Lipschitz, u_{|_{E_{0}}} = 0, u_{|_{E_{1}}} = 1} and ∇u is the gradient of u . cap(E_{0}, E_{1}, D) = cap_{n}(E_{0}, E_{1}, D) is called the conformal capacity of the condenser (E_{0}, E_{1}, D). In this communication I establish some inequalities between the p-module and the p-capacity and also that

M_{p}Γ(F_{0}, F_{1}, D) = cap_{p}(F_{0}, F_{1}, D) (p > 1)

if F_{0}, F_{1} ⊂ D̄ ⊂ \overline{R^{n}}, d(F_{0}, F_{1}) > 0 and ∀ξ ∈ ∂D ∩ F_{i}, for i = 0 or i = 1,

lim_{x→ξ,x∈D} inf_{γ} H^{1} [γ(F_{i}, x)] = 0 ,

the infimum being taken over all arcs γ = γ(F_{i}, x) joining F; and x in D (i = 0, 1).