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

Abstract

We establish an optimal L p -regularity theory for solutions to fourth order elliptic systems with antisymmetric potentials in all supercritical dimensions n ≥ 5:where and D, E, Ω, G satisfy the growth condition (GC-4), under the smallness condition of a critical scale invariant norm of ∇u and ∇ 2 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 p -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.

Global conservation law for an even order elliptic system with antisymmetric potential

“…Furthermore, in [23,9,10], more than the Hölder continuity, an optimal L p theory for weak solutions to (inhomogenous version of) (1.5), (1.7) and (1.10) was established. There are also alternative ways to derive Hölder continuity (or L p theory) of (1.5), (1.7) and (1.10); we refer the readers to [22,25,8,6] for more information.…”