Necessary and sufficient conditions of compactness of certain embeddings of Sobolev spaces

Abstract

Necessary and sufficient conditions on an open set Ω ⊂ ℝⁿ are obtained ensuring that for l, m ∈ ℕ₀, m < l the embedding Wᵢₙᵢₜ(Ω) ⊂ Wₘ(Ω) is compact, where Wₘ(Ω) is the Sobolev space and Wᵢₙᵢₜ(Ω) is the closure in Wᵢₙᵢₜ(Ω) of the space of all infinitely continuously differentiable functions on Ω with supports compact in Ω.