Energy identity for \(m\)-harmonic maps. (English) Zbl 1037.58011

From the text: For \(m\geq 3\), let \(M\), with dimension \(m\), and \(N\subset\mathbb{R}^n\) be compact Riemannian manifolds without boundaries. Let \(W^{1,m}(M,N)\) be defined by \[ W^{1,m}(M,N)\equiv\bigl\{v\in W^{1,m}(M,\mathbb{R}^k): v(x) \in N\text{ a.e. }x \in M\bigr\}. \] We prove the energy identity (*) for a sequence of weakly convergent \(m\)-harmonic maps in \(C^1(M,N)\):
Theorem A. For \(m\geq 3\), assume that \(\{u_n\}\subset C^1(M,N)\) is a sequence of \(m\)-harmonic maps which converges to \(u\in W^{1,m}(M,N)\) weakly. Then \(u\in C^1 (M,N)\) is an \(m\)-harmonic map and there are a nonnegative integer \(l\) depending only on \(M,N\) and nonconstant \(m\)-harmonic maps \(\{\omega_i\}^l_{i=1} \subset C^1(S^m,N)\) \((S^m=\mathbb{R}^m\cup\{\infty\})\) such that \[ \lim_{n\to\infty} \int_M | Du_n|^m=\int_M| Du|^m+ \sum^l_{i=1} \int_{S^m} | D \omega_i |^m.\tag{*} \] We also generalize the result to certain regular approximated \(m\)-harmonic maps whose tension fields are bounded in \(L^{\frac {m}{m-1}}\).


58E20 Harmonic maps, etc.