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

Abstract: Let $1 \leq q \le 2$ and denote by $2 \leq q&#39;$ its corresponding conjugate exponent. We prove the continuity of solutions $u \in W^{{1,(\frac{n}{n-1},q&#39;)}(B^n,} \mathbb{R}<sup>m)$ to the critical elliptic system $-Δu = Ω\cdot \nabla u$ in dimension $n \ge 3$, where the potential $$Ω\in L^{{(n,q)}(B^n,} \mathfrak{so}(m) \otimes \wedge<sup>1)$$ is antisymmetric. First, we construct $$P \in W^{{1,(n,q)}(B^n,} \mathrm{SO}(m))$$ such that the PDE can be rewritten as $$-\operatorname{div}(P^{-1}du) = \ast dξ\cdot P<sup>{-1}du$$, 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. $$-\operatorname{div}(Adu)=d^\ast B \cdot du.$$