Boundary regularity for weak heat flows (Q1600025)

The partial regularity of the weak flow of harmonic maps from a Riemannian manifold \(M\) with boundary into a general compact Riemannian manifold without boundary is considered. It is assumed that the weak flow obeys a monotonicity inequality (in the spirit of Struwe's, though with modifications at the boundary) and a boundary energy inequality, both of which hold for smooth solutions. Both conditions are formulated only for the case that \(M\) is a Euclidean half space. Under these assumptions, it is proved that the flow is regular outside a closed singular set. The singular set has parabolic Hausdorff measure (w.r.t.\ the dimension of \(M\)) zero in the closure of \(M\times R\). The article reviewed here discusses only the boundary regularity. For interior regularity, the reader is referred to another article by the same author: [\textit{X. G. Liu}, Partial regularity for weak heat flow into general compact Riemannian manifold. Preprint]. The proof of the theorem uses Hardy-BMO duality, a well-established tool in harmonic map theory, in order to make proper use of the special structure of the equation, which is more obvious in the cases of the symmetric targets that have been considered so far.