Density properties for fractional Sobolev spaces

Abstract

Abstract. Aim of this paper is to give the details of the proof of some density properties of smooth and compactly supported functions in the fractional Sobolev spaces and suitable modifications of them, which have recently found application in variational problems. The arguments are rather technical, but, roughly speaking, they rely on a basic technique of convolution (which makes functions C ∞ ), joined with a cut-off (which makes their support compact), with some care needed in order not to exceed the original support.

Fractional Laplacian equations with critical Sobolev exponent

“…Notice that this functional is well defined thanks to the definition of X 0 , to [19,Lemmas 5,6,9 and footnote 4], [11,Theorem 6] and to the fact that is bounded. Moreover, J K , λ is Fréchet differentiable in u ∈ X 0 and for any ϕ ∈ X 0…”

Realization of the fractional Laplacian with nonlocal exterior conditions via forms method

“…We have the following result. [19] (Ω). Notice also that W s,2 0 (Ω) = W s,2 (Ω) for every 0 < s ≤ 1 2 (see e.g.…”

“…[8,24,37]). For more information on fractional order Sobolev spaces we refer to [14,19,24,37]. Next, let β ∈ L 1 (R n \ Ω) be fixed and define the fractional order Sobolev type space…”

Properties of eigenvalues and some regularities on fractionalp-Laplacian with singular weights

“…The space W s,p 0 (Ω) is a separable and uniformly convex Banach space and it can be defined as the completion of C ∞ c (Ω) with respect to the norm (2.1). It is well-known that W s,p [3,11,14,16]). We shall also include the p-Laplacian case and the solution space in this case is the usual Sobolev space W 1,p 0 (Ω) endowed with the norm u = ´Ω |∇u| p dx 1/p .…”