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

摘要

Let Omega subset of R-n (n greater than or equal to 2) be open and N subset of R-K a smooth, compact Riemannian manifold without boundary. We consider the approximated harmonic map equation Deltau + A (u) (delu, delu) = f for maps u is an element of H-1(Omega, N), where f subset of L-p (Omega, R-K). For p > n/2, we prove Holder continuity for weak solutions which satisfy a certain smallness condition. For p = 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 less than or equal to 4. [References: 24]