Density of Lipschitz mappings in the class of Sobolev mappings between metric spaces (Q1006803)

Let \((X,d,\mu)\) be a metric-measure space equipped with a finite doubling measure. The space \((X,d,\mu)\) is said to support the \(p\)-Poincaré inequality for some \(p\in[1,\infty)\) if \(\mu\) is a doubling measure and there exist constants \(C_p >0\) and \(\lambda\geq1\) such that for every ball \(B\subset X\), for every integrable function \(f\in L^1(\lambda B)\) and for every \(0\leq g\in L^p (\lambda B)\) being a \(p\)-weak upper gradient of \(f\) on \(\lambda B\), the inequality \[ \frac{1}{\mu(B)}\int_B |f-f_B|\,d\mu\leq C_p\,({\mathrm{\,diam\,}}B) \left(\frac{1} {\mu(\lambda B)}\int_{\lambda B} g^p \,d\mu\right)^{1/p} \] is satisfied, where \(f_B=\frac{1} {\mu(B)}\int_B f \,d\mu\). Also, for \(p\in[1,\infty)\), any metric-measure space \((X,d,\mu)\) and metric space \((Y,d_Y)\), the mapping \(F\in L^p (X,Y)\) is said to belong to the Newtonian-Sobolev class of mappings \(N^{1,p}(X,Y)\) if there is a Borel function \(0\leq g\in L^p (X)\) such that \[ d_Y (F(\gamma(a)),F(\gamma(b)))\leq\int_{\gamma}g \] for every rectifiable curve \(\gamma : [a,b]\rightarrow X\). In this paper, the author proves that Lipschitz mappings are dense in the Newton--Sobolev classes \(N^{1,p}(X,\,Y)\) of mappings from spaces \(X\) supporting \(p\)-Poincaré inequalities into a finite Lipschitz polyhedron \(Y\) if and only if \(Y\) is \([p]\)-connected, \(\pi_1 (Y)=\pi_2 (Y)=\cdots=\pi_{[p]}(Y)=0\), where \(p\in[1,\infty)\) and \([p]\) denotes the largest integer less than or equal to \(p\).