\(p\)-Dirichlet energy minimizing maps into a complete manifold
Let \(M\) be a compact \(m\)-dimensional \(C^2\) Riemannian submanifold of \(\mathbb{R}^s\) with (or without) boundary \(\partial M\) and \(N\) an \(n\)-dimensional \(C^2\) complete connected Riemannian submanifold without boundary of some Euclidean space \(\mathbb{R}^k\). The \(p\)-Dirichlet energy functional is the \(L^p\)-norm of the gradient defined on the admissible mapping space \(L^{1,p} (M,N) = \{v \in L^{1,p} (M,\mathbb{R}^k) : v(x) \in N\) for \({\mathcal L}^m\) a.e. \(x \in M\}\), where \({\mathcal L}^m\) is the \(m\)-dimensional Hausdorff measure induced by the metric of \(M\) and \(1 < p \leq m\). The map \(u \in L^{1,p} (M,N)\) is called \(p\)-Dirichlet energy minimizing if \(\int_M |\nabla u|^p \leq \int_M |\nabla v|^p\) for all \(v\in L^{1,p} (M,N)\) in the same (relative) homotopy class of \(u\). Existence theorems of \(p\)-Dirichlet energy minimizing maps are presented. Propositions about the partial regularity of a \(p\)-Dirichlet energy minimizing map are proved.
scientific article; zbMATH DE number 902585