Weak and strong density results for the Dirichlet energy (Q1432719)

Summary: Let \({\mathcal Y}\) be a smooth oriented Riemannian manifold which is compact, connected, without boundary and with second homology group without torsion. In this paper we characterize the sequential weak closure of smooth graphs in \(B^n\times{\mathcal Y}\) with equibounded Dirichlet energies, \(B^n\) being the unit ball in \(\mathbb{R}^n\). More precisely, weak limits of graphs of smooth maps \(u_k:B^n\to {\mathcal Y}\) with equibounded Dirichlet integral give rise to elements of the space \(\text{cart}^{2,1} (B^n\times{\mathcal Y})\). In this paper we prove that every element \(T\) in \(\text{cart}^{2,1} (B^n\times{\mathcal Y})\) is the weak limit of a sequence \(\{u_k\}\) of smooth graphs with equibounded Dirichlet energies. Moreover, in dimension \(n=2\), we show that the sequence \(\{u_k\}\) can be chosen in such a way that the energy of \(u_k\) converges to the energy of \(T\).