## 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 $$L^{\frac {m}{m-1}}$$.

### MSC:

 5.8e+21 Harmonic maps, etc.

### Keywords:

$$p$$-harmonic maps; energy identity