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

Research paper by Roger Moser

Indexed on: 01 Feb '03Published on: 01 Feb '03Published in: Mathematische Zeitschrift

#### 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 $$n \le 4$$.