具反对称势的临界n-拉普拉斯方程组的不连续解
A Discontinuous Solution of the Critical n-Laplace System with Antisymmetric Potential
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中文总结 AI 辅助
针对Rivière提出的正则性问题，构造了具反对称势的临界n-拉普拉斯方程组的不连续解，给出其否定回答，同时明确了相关空间正则性结果。
中文摘要 AI 辅助
设n>2，我们构造了一个映射U∈W^{1,n}(B^n,ℝ^{n+2})，它在原点处不连续，在去心球B^n\{0}上光滑，同时构造了一个反对称势Ω∈L^n(B^n,so(n+2)⊗ℝ^n)，使得在分布意义下D'(B^n)中满足- Div(|∇U|^{n-2}∇U)=Ω·|∇U|^{n-2}∇U。这对Rivière提出的正则性问题给出了否定回答。我们的势具有洛伦兹空间正则性Ω∈∩_{q>2}L^{(n,q)}\L^{(n,2)}；此外，对给定的1<p<∞，我们可使∇U∈L^{(n,p)}但∇U∉L^{(n,1)}。该构造并未给出弱n-调和映射或高维H-系统正则性的反例。该示例由ChatGPT 5.6 Sol于2026年8月5日生成，研究工作由作者撰写并经彻底审核以确保正确性。
英文摘要
Let $n>2$. We construct a map $U\in W^{1,n}(B^n,\mathbb{R}^{n+2})$ that is discontinuous at the origin and smooth on the punctured ball $B^n \setminus \{0\}$, together with an antisymmetric potential $Ω\in L^n(B^n,so(n+2)\otimes\mathbb{R}^n)$ such that $-\mathrm{Div}(|\nabla U|^{n-2}\nabla U)=Ω\cdot |\nabla U|^{n-2}\nabla U$ in $D'(B^n)$. This gives a negative answer to a regularity question posed by Rivière. Our potential admits the Lorentz-space regularity $Ω\in \bigcap_{q>2}L^{(n,q)} \setminus L^{(n,2)}$. In addition for given $1<p<\infty$ we can enforce $\nabla U \in L^{(n,p)}$ but $\nabla U \notin L^{(n,1)}$. The construction does not give a counterexample to regularity for weakly $n$-harmonic maps or for higher-dimensional $H$-systems. The example was generated by ChatGPT 5.6 Sol on August 5, 2026. The work itself was written by the author and thoroughly reviewed to ensure its correctness.
