Sobolev embeddings and distance functions

For every α > 0 and x 1 ∈ R , we set

Then we consider the quasibounded open set

Observe that, for this set, we have d Ω α γ ∉ L 1 ⁢ ( Ω α ) for any 0 < γ < ∞ . Thus, by Theorem 5.1, part (i), we have

On the other hand, since Ω α is bounded in the x 2 direction, we easily get that λ 2 ⁢ ( Ω α ) > 0 , that is

As for the compactness of this embedding, we observe that this cannot be directly inferred from Theorem 5.4 since we are in the critical situation p = 2 = N . Nevertheless, we are going to show that actually such an embedding is compact if 𝛼 is large enough, thanks to the peculiar geometry of the set Ω α . In particular, the Dirichlet-Laplacian on Ω α has a discrete spectrum.

We define

We denote by w Ω α the torsion function of Ω α , defined as

where w Ω α , R ∈ W 0 1 , 2 ⁢ ( Ω α , R ) is the torsion function of the bounded set Ω α , R , i.e. it solves

(see [12, Definition 2.2]). We observe that w Ω α is a bounded function, by [12, Theorem 1.3]. In order to prove the compactness of the embedding of D 0 1 , 2 ⁢ ( Ω α ) ↪ L 2 ⁢ ( Ω α ) , it is sufficient to prove that

thanks to [12, Theorem 1.3]. We will achieve (A.1) by exploiting the geometry of Ω α in order to construct a suitable upper barrier. For every α > 0 and x 1 ∈ R , we set

Observe that F 1 has a bounded second order derivative, i.e. there exists L > 0 such that

Thus, if we take α > 0 such that α 2 > L , we obtain

where

With such a choice of 𝛼, we consider the function

We claim that this is the desired upper barrier. Indeed, by construction we have U α ≥ 0 and, thanks to (A.2), it holds

By applying the comparison principle in every Ω α , R , we get that

and such an estimate does not depend on 𝑅. Hence, by sending 𝑅 to ∞, we have that

By using the properties of F α = ( f α ) 2 and the previous estimate, we eventually get (A.1).