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

TL;DR

This paper develops an optimal $L^p$-regularity theory for fourth order elliptic systems with antisymmetric potentials in higher dimensions, improving regularity results and extending previous theories to new classes of systems.

Contribution

It introduces a new $L^p$-regularity framework for fourth order elliptic systems with antisymmetric potentials, applicable in all supercritical dimensions, and extends harmonic map regularity theory.

Findings

Improves Struwe's Hölder regularity to any exponent in (0,1) when $f=0$.

Extends $L^p$-regularity theory to biharmonic maps and heat flow.

Confirms the extension of harmonic map regularity to 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 $n\ge 5$: $$ \Delta^2 u=\Delta(D\cdot\nabla u)+div(E\cdot\nabla u)+(\Delta\Omega+G)\cdot\nabla u +f \qquad \ {\rm{in}}\ B^n, $$ 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-Rivi\`ere, Struwe, and Wang. In particular, our results improve Struwe's H\"older regularity theorem to any H\"older 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 Rivi\`ere-Struwe's second order elliptic systems with antisymmetric potentials under the growth condition (GC-2) in all dimensions, which confirms an expectation by Sharp.