Energy identity for a class of approximate biharmonic maps into sphere in dimension four

Abstract

We consdier in dimension four weakly convergent sequences of approximate biharmonic maos into sphere with bi-tension fields bounded in $L^p$ for some $p>1$. We prove an energy identity that accounts for the loss of Hessian energies by the sum of Hessian energies over finitely many nontrivial biharmonic maps on $\mathbb R^4$.