We prove that every polyharmonic map u ∈ Wm, 2 (Bn, SN - 1) is smooth in the critical dimension n = 2 m. 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. © 2009 Elsevier Masson SAS. All rights reserved.
Goldstein, P., Strzelecki, P., & Zatorska-Goldstein, A. (2009). On polyharmonic maps into spheres in the critical dimension. Annales de l’Institut Henri Poincare (C) Analyse Non Lineaire, 26(4), 1387–1405. https://doi.org/10.1016/j.anihpc.2008.10.008