Regularity of generalized sphere valued p - harmonic maps with small mean oscillations

We prove (see Theorem 1.3 below) that a generalized harmonic map into a round sphere, i.e. a map u ∈ W 1,1 loc ( , Sn−1) which solves the system div (ui∇uj − uj∇ui) = 0, i, j = 1, . . . , n, is smooth as soon as |∇u| ∈ L for any q > 1, and the norm of u in BMO is sufficiently small. Here, ⊂ R is open, and m, n are arbitrary. This extends various earlier results of Almeida [1], Ge [15], and R. Moser [38]. A version of this result for generalized p-harmonic maps into spheres is also proved. The proofs rely on the duality of Hardy space and BMO combined with L stability of the Hodge decomposition and reverse Hölder inequalities. The results of this note belong to the regularity theory of nonlinear elliptic systems with the right hand side growing critically with the gradient of solution. The particular system we have in mind may appear special but it is closely linked with vast parts of geometric analysis and PDE. In case m = n = 2 its solutions agree with asymptotic limits of solutions of a complex valued Ginzburg–Landau equation involving a small parameter, see Bethuel, Brezis and Hélein [4] and Hardt and Lin [22]. We consider solutions with weak integrability assumptions, weaker than those which are needed for a variational interpretration of the system. The proofs are based on a mixture of (well known) subtle hard analytic methods.