Sobolev spaces and trace theorem on the Sierpinski gasket

2019/03/17 by Cao, Shiping, Qiu, Hua
#28A80 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1903.07012

Abstract

On the Sierpinski gasket SG, we consider Sobolev spaces L²_σ(SG) associated with the standard Laplacian Δ with order σ≥ 0. When σ∈ℤ⁺, L²_σ(SG) consists of functions equipped with L² norms of the function itself and its Laplacians up to σ order; when σ∉ ℤ⁺, we fill up the gaps between integer orders by using complex interpolation. Let L²_{σ, D}(SG)=(I-Δ_D)^-σL²(SG) where Δ_D is the Dirichlet Laplacian associated with Δ. Let \p_n\_{n≥ 0} be a collection of countably many points located along one of the symmetrical axes of SG. We make a full characterization of the trace spaces of L²_σ(SG) and L²_σ,D(SG) to \p_n\_{n≥ 0}. Using this, we get a full description of the relationship between L²_σ(SG) and L²_σ,D(SG) for σ≥ 0. The result indicates that when σ-(log 3)/(2log5)∈ ℤ⁺, L²_{σ, D}(SG) is not closed in L²_σ(SG) and has an infinite codimension. Otherwise, L²_{σ, D}(SG) is closed in L²_σ(SG) with a finite codimension. Similar result holds for the Neumann case. Another consequence of the trace result is that the Sobolev spaces L²_σ(SG) are stable under complex interpolation for σ≥ 0 although they are defined by piecewise interpolation between integer orders.

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