AI 中文总结
该研究针对n≥3维带反对称势的临界椭圆型方程组，通过构造SO(m)取值的映射与Rivière的GL(m)-规范，建立(A,B)-守恒律，证明了解的连续性。
AI 中文摘要
设1 ≤ q ≤ 2，记q'为其对应的共轭指数。我们证明了n ≥ 3维下临界椭圆型方程组 -Δu = Ω·∇u 的解u ∈ W^{1,(\frac{n}{n-1},q')}(B^n, ℝ^m) 的连续性，其中势Ω ∈ L^{(n,q)}(B^n, 𝔰𝔬(m) ⊗ ∧^1) 是反对称的。首先，我们构造P ∈ W^{1,(n,q)}(B^n, SO(m))，使得该偏微分方程可重写为 -div(P^{-1}du) = *dξ·P^{-1}du，该结构在旋转P的意义下几乎是雅可比结构。其次，我们给出Rivière的GL(m)-规范以建立“完整的”(A,B)-守恒律，即 -div(Adu)=d^*B·du。
英文摘要
Let $1 \leq q \le 2$ and denote by $2 \leq q'$ its corresponding conjugate exponent. We prove the continuity of solutions $u \in W^{1,(\frac{n}{n-1},q')}(B^n, \mathbb{R}^m)$ to the critical elliptic system $-Δu = Ω\cdot \nabla u$ in dimension $n \ge 3$, where the potential $Ω\in L^{(n,q)}(B^n, \mathfrak{so}(m) \otimes \wedge^1)$ is antisymmetric. First, we construct $P \in W^{1,(n,q)}(B^n, \mathrm{SO}(m))$ such that the PDE can be rewritten as $-\operatorname{div}(P^{-1}du) = \ast dξ\cdot P^{-1}du$, 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. $-\operatorname{div}(Adu)=d^\ast B \cdot du$. We show that the assumption on $Ω\in L^{(n,q)} (B^n, \mathfrak{so}(m) \otimes \wedge^1) $ for $q\leq 2$ is optimal.
