Sobolev spaces and harmonic maps for metric space targets

When one studies variational problems for maps between Riemannian manifolds one must consider spaces which we denote Vr'(r2,X). Here ft is a compact domain in a Riemannian manifold, X is a second Riemannian manifold, p G [l,oo), and W indicates that the first derivatives of the map are L(0). For p > n such maps will be continuous, and the corresponding space W(Cl^X) can be given the structure of a smooth Banach manifold. This is because, for p > n, any map which is close in W^ distance to a map ^o can be described as a pointwise small deformation of UQ. This linear space of W deformations is then a Banach space on which one can locally model W'(Q^ X). For p < n this is no longer possible, and the definition of the space W>{p,,X) becomes much less clear. This problem was first encountered by C.B. Morrey [Mo] in case n = dimfi = 2 and p = 2. A great deal of effort was spent by Morrey to give a definition of this space. In more recent times people have exploited the embedding theorem of J. Nash, and considered X to be a smooth submanifold of a Euclidean space M^. If we define W'(fi, X) to be the subset of the Banach space VF^f^R^) consisting of those maps with image essentially in X, it turns out that this gives a workable definition for many purposes. An aesthetic drawback of this definition is that the space VF'(J7, X) should depend only on the metric of X and not on the embedding of X into R. A much more serious difficulty arises if one attempts to consider maps to spaces X which are not smooth Riemannian manifolds. These