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 $2m$.