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

Abstract.

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 $n \le 4$.