Regularity for the approximated harmonic map equation and application to the heat flow for harmonic maps

摘要

Let $Omega subset{mathbb R}^n (n ge 2)$ be open and $N subset {mathbb R}^K$ a smooth, compact Riemannian manifold without boundary. We consider the approximated harmonic map equation $Delta u + A(u)(nabla u, nabla u) = f$ for maps $u in {H^1(Omega, N)}$ , where $f in L^p(Omega,{mathbb R}^K)$ . For $p > frac{n}{2}$ , we prove Hölder continuity for weak solution s which satisfy a certain smallness condition. For $p = frac{n}{2}$ , we derive an energy estimate which allows to prove partial regularity for stationary solutions of the heat flow for harmonic maps in dimension $n le 4$ .