$L^p$-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 $n\ge 5$: $${\Delta }^{2}u=\Delta (D\cdot \nabla u)+div(E\cdot \nabla u)+(\Delta \Omega +G)\cdot \nabla u+f\mathrm{in}{B}^n,$$ where $\Omega \in W^{1,2}(B^n,s{o}_m)$ 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 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.

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