2 - A density result for fractional Sobolev spaces

Summary

The aim of this chapter is to give the details of the proof of some density properties of smooth and compactly supported functions in fractional Sobolev spaces and suitable modifications thereof, which have recently found application in variational problems. The arguments are rather technical, but roughly speaking, they rely on a basic technique of convolution, joined with a cutoff, with some care needed not to exceed the original support.

As is wellknown, the natural functional space associated with the classical Laplace setting is the Sobolev space H₀¹ (Ω). When ∂Ω is sufficiently regular, such a space may be equivalently defined as the set of functions that belongs to L²(Ω), together with their weak derivatives of order 1 and that vanish along ∂Ω in the trace sense, or as the closure of C₀^∞ (Ω) in the norm of H¹(Ω) (see, e.g., [3]).

In this chapter, we study the density properties of the fractional space X₀^s,p (Ω) (see Section 2.1 for a definition) in order to establish a result similar to the one known in the classical case. Precisely, we investigate the relation between the spaces X₀^s,p (Ω) and C₀^∞ (Ω).

Sometimes density properties in the fractional framework are similar to those in the classical case, as happens for the results presented in this chapter. The reader should not infer that such fractional density properties are always well expected. For instance, it turns out that fractional harmonic functions are locally dense in the space of continuous and smooth functions (see [86]). This is in clear contrast with the classical case because harmonic functions can never approximate a function with a strict maximum (simply by the maximum principle). The results of this chapter are based on the paper [103].

The main theorems

Let K: ℝⁿ \ ﹛0﹜→(0,+∞) satisfy assumptions (1.55) and (1.56), and let X^s,p(ℝⁿ) be the linear space of Lebesgue measurable functions u from ℝⁿ to ℝ such that

Then X₀^s,p (Ω) denotes the space of functions u ∈ X^s,p(ℝⁿ) that vanish a.e. in. Clearly, the quantity in (2.1) defines a norm on X^s,p(ℝⁿ) and X₀^s,p (Ω): such a norm will be denoted in this chapter by ‖ · ‖.