Comment. Math. Helvetici 70 (1995) 161 - 169 0010-2571/95/010161-0951.50 + 0.20/0 9 1995 Birkh/iuser Verlag, Basel Remarks on approximate harmonic maps YUMNEI CHEN* AND FANG HUA LIN** w Introduction The analytical difficulties in the study of harmonic maps come from the fact that the maps take their values in a curved, compact Riemannian manifold N. One natural way to tackle such a problem is to use a so called penalty approximation, that is, to relax this nonlinear, nonconvex constraint. Roughly speaking, one studies, instead of the standard Dirichlet integrals, the following variational integral where M is a compact, Riemann manifold with (or without boundary) OM, and U'M---,~k Here we view, via Nash's isometric embedding, N as a compact submanifold of R k, and d(U, N) denotes the distance from U to N. The above approach has been employed successfully by Chen and Struwe [CS] in establishing the global existence of weak solutions to the heat flow of harmonic maps. Moreover, to study such approximate energy functional (1.1) may also be natural in the Ginzburg-Laudau's approach to various physical problems, see, e.g., [BBH] and references therein. The present note is bought out by our previous work [CL] on the evolution of harmonic maps with Dirichlet boundary conditions. We shall establish here first the Schoen-Uhlenbeck's Theorem, or "small energy regularity theorem", for energy minimizing maps. The problem is essentially reduces to obtain an a priori estimate for a family of smooth approximate solutions with small energy. As an application of our method we shall also prove one of the main results of [BBH2] concerning asymptotic limits for the Ginzburg-Landau model of scalar fields. * The research is partially supported by the NSF-grant DMS #9123532. 7" The research is partially supported by the NSF-grant DMS #9149555. 161