Sobolev spaces and trace theorem on the Sierpinski gasket

Abstract

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