Liouville-type theorems of \(p\)-harmonic maps with free boundary values (Q633867)

The author obtains some Liouville-type theorems for \(p\)-harmonic maps with free boundary. Let \({\mathbb R}^m_+\) \((m\geq 3)\) be the upper-half space with boundary and \(g_0\) the standard flat metric on it. Let \((N^n, h)\) be a Riemannian \(n\)-manifold with \(n \geq 2\) and let \(S\) be a submanifold of \(N\) of dimension \(d\), \(1\leq d \leq n-1\). Let \(u :({\mathbb R}^m_+, g_0) \to (N, h)\) be a \(C^1\) \(p\)-harmonic map satisfying \(u(\partial{{\mathbb R}^m_+}) \subset S \subset N\) and \(\frac{\partial u}{\partial \nu}(x) \perp T_{u(x)}S\) for any \(x \in \partial {\mathbb R}^m_+\), where \(\nu\) is the unit normal vector field to \(\partial {\mathbb R}^m_+\) and \(p \geq 2\). Under these assumptions, the author proves that if the \(p\)-energy satisfies \(E_p(u) <\infty\), then \(u\) must be a constant map. The author also shows that the same result holds if the condition \(E_p(u) <\infty\) is replaced by the asymptotic behavior of the map \(u\) at infinity, \(u(x) \to Q_0\) (a point) as \(|x|\to \infty\).