On polyharmonic maps into spheres in the critical dimension

Abstract

We prove that every polyharmonic map u\in W^{m,2}(\mathbb{B}^{n},\mathbb{S}^{N−1}) is smooth in the critical dimension n = 2m . Moreover, in every dimension n , a weak limit u\in W^{m,2}(\mathbb{B}^{n},\mathbb{S}^{N−1}) of a sequence of polyharmonic maps u_{j}\in W^{m,2}(\mathbb{B}^{n},\mathbb{S}^{N−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 .