$$L^p$$-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 $$n\ge 5$$ : $$\begin{aligned} \Delta ^2 u=\Delta (D\cdot \nabla u)+\text {div}(E\cdot \nabla u) +(\Delta \Omega +G)\cdot \nabla u +f \qquad \ \mathrm{{in}}\ B^n, \end{aligned}$$ where $$\Omega \in W^{1,2}(B^n, so_m)$$ is antisymmetric and $$f\in L^p(B^n)$$ , and $$D, E, \Omega , G$$ satisfy the growth condition (GC-4), under the smallness condition of a critical scale invariant norm of $$\nabla u$$ and $$\nabla ^2 u$$ . 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 $$\alpha \in (0,1)$$ when $$f\equiv 0$$ , and have applications to both approximate biharmonic maps and heat flow of biharmonic maps. As a by-product of our techniques, we also partially extend the $$L^p$$ -regularity theory of approximate harmonic maps by Moser to Rivière-Struwe’s second order elliptic systems with antisymmetric potentials under the growth condition (GC-2) in all dimensions $$n\ge 2$$ when $$p>\frac{n}{2}$$ , which partially confirms an interesting expectation by Sharp.