$L^p$-regularity for fourth order elliptic systems with antisymmetric potentials in higher dimensions

We establish an optimal L-regularity theory for solutions to fourth order elliptic systems with antisymmetric potentials in all supercritical dimensions n ≥ 5: ∆u = ∆(D · ∇u) + div(E · ∇u) + (∆Ω +G) · ∇u+ f in B, where Ω ∈ W (B, som) is antisymmetric and f ∈ L (B), and D,E,Ω, G satisfy the growth condition (GC-4), under the smallness condition of a critical scale invariant norm of ∇u and ∇u. This system was brought into lights from the study of regularity of (stationary) biharmonic maps between manifolds by Lamm-Rivière, Struwe, and Wang. In particular, our results improve Struwe’s Hölder regularity theorem to any Hölder exponent α ∈ (0, 1) when f ≡ 0, and have applications to both approximate biharmonic maps and heat flow of biharmonic maps. As a by-product of the techniques, we also extend the L-regularity theory of harmonic maps by Moser to Rivière-Struwe’s second order elliptic systems with antisymmetric potentials under the growth condition (GC-2) in all dimensions, which confirms an expectation by Sharp.