On polyharmonic maps into spheres in the critical dimension

We prove that every polyharmonic map u∈Wm,2(Bn,SN−1) is smooth in the critical dimension n=2m. Moreover, in every dimension n, a weak limit u∈Wm,2(Bn,SN−1) of a sequence of polyharmonic maps uj∈Wm,2(Bn,SN−1) is also polyharmonic.The proofs are based on the equivalence of the polyharmonic map equations with a system of lower order conservation laws in divergence-like form. The proof of regularity in dimension 2m uses estimates by Riesz potentials and Sobolev inequalities; it can be generalized to a wide class of nonlinear elliptic systems of order 2m.