Removable sets for Sobolev spaces
Let \(\Omega\subset{\mathbb R}^n\) \((n\geq 2)\) be an open set. For \(1\leq p<\infty\), the Sobolev space \(W^{1,p}(\Omega)\) consists of all \(u\in L^p(\Omega)\) such that there exist \(\partial_j u\in L^p(\Omega)\) \((1\leq j\leq n)\) for which \[ \int_{\Omega} u \partial_j\psi\, dx = -\int_{\Omega} \psi \partial_j u\, dx \quad (\psi\in C_0^1 (\Omega), \; 1\leq j\leq n). \] A closed set \(E\subset {\mathbb R}^n\) of zero \(n\)-dimensional Lebesgue measure is said to be removable for \(W^{1,p}\) if the algebraic identity \(W^{1,p}({\mathbb R}^n\setminus E)=W^{1,p}({\mathbb R}^n)\) holds. On the other hand, a set \(E\subset {\mathbb R}^{n-1}\) is called \(p\)-porous \((1<p<n-1)\) if for a.e. \(x\in E\) with respect to \((n-1)\)-dimensional Hausdorff measure there exists a sequence \(\{r_i\}_{i=1}^{\infty}\) and a constant \(C_x\) such that \(r_i\rightarrow 0\) as \(i\rightarrow \infty\), and for each \((n-1)\)-dimensional ball \(B(x,r_i)\), a ball \(B_i\subset B(x,r_i)\setminus E\) of radius not less than \(C_x r_i^{(n-1)/(n-p)}\). For \(n-1\leq p<n\), \(p\)-porous sets are defined by replacing the balls \(B_i\) by continua \(F_i\) of diameters not less than \(C_x r_i^{(n-1)/(n-p)}\). Finally, \(E\) is \(n\)-porous if the diameter of \(F_i\) is not less than \(C_x r_i\exp(-1/C_x r_i)\). The main result of the paper establishes that if \(E\subset {\mathbb R}^{n-1}\) is \(p\)-porous (\(1<p\leq n\)), then \(E\) is removable for \(W^{1,p}\). From this, a consequence about the extendability of Sobolev functions is derived. Also, a characterization of the sets \(E\) which are removable for \(W^{1,p}\) \((1<p<\infty)\) as those such that \({\mathbb R}^n\setminus E\) supports a so-called \(p\)-Poincaré inequality is given.
scientific article; zbMATH DE number 2062431
scientific article