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 Holder 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 \(n \le 4\).