The Sobolev Problem

Abstract

Let G ⊂ R ⁿ be a bounded domain, and let âG = Γ₀ ∪ Γ₁ ∪... Γk ∈^∞ be the boundary of G. Assume that Γ₀ denotes an (n — 1)-dimensional compact set_that is the exterior boundary of the domain G. Denote by Γ_j (j = 1, ..., k̄) the i_j-dimensional manifold without boundary lying inside of Γ_o, 0≤ i_j ≤ n — 1. Let ί = n - i_j denotes the codimensionality of Γ_j. Assume that Γ_j ∈ C∞ (j = 0, ...,k̄), and Γ_j ∩ Γ_k =Ø for j≠k.