$L^p$-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 $n\ge 5$: $$ Δ^2 u=Δ(D\cdot\nabla u)+div(E\cdot\nabla u)+(ΔΩ+G)\cdot\nabla u +f \qquad \ {\rm{in}}\ B^n, $$ where $Ω\in W^{1,2}(B^n, so_m)$ is antisymmetric and $f\in L^p(B^n)$, and $D, E, Ω, G$ 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 $α\in (0,1)$ 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.

Theorems & Definitions (35)

• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Corollary 1.5
• Corollary 1.6
• Theorem 1.7
• Theorem 1.8
• proof : Proof of Theorem \ref{['thm:second order']}
• Proposition 2.1
• Proposition 2.2