Regularity and Rivière's $GL(m)$-Gauge Construction for Elliptic Systems with Antisymmetric Potentials in Arbitrary Dimensions

Abstract

Let $1\le q\le 2$ and denote by $2\le q^{\prime }$ its corresponding conjugate exponent. We prove the continuity of solutions $u\in W^{1,(\frac{n}{n-1},q^{\prime })}(B^n,{\mathbb{R}}^m)$ to the critical elliptic system $-\Delta u=\Omega \cdot \nabla u$ in dimension $n\ge 3$, where the potential $\Omega \in L^{(n,q)}(B^n,\mathfrak{so}(m)\otimes {\wedge }^1)$ is antisymmetric. First, we construct $P\in W^{1,(n,q)}(B^n,\mathrm{SO}(m))$ such that the PDE can be rewritten as $-\mathrm{div}(P^{-1}du)=\ast d\xi \cdot P^{-1}du$, which is nearly a Jacobian structure up to the rotation $P$. Second, we provide a Rivière's $\mathrm{GL}(m)$-Gauge in order to establish a "full" $(A,B)$-conservation law, i.e. $-\mathrm{div}(Adu)=d^{\ast }B\cdot du$. We show that the assumption on $\Omega \in L^{(n,q)}(B^n,\mathfrak{so}(m)\otimes {\wedge }^1)$ for $q\le 2$ is optimal.