Carnot群上的Korevaar-Schoen能量与分数阶Sobolev映射的插值
Korevaar-Schoen Energy and Interpolations of Fractional Sobolev mappings on Carnot groups
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中文总结 AI 辅助
该研究将欧氏空间的弱接触方程结果推广到次黎曼框架下的Carnot群，探究分数阶Sobolev映射的Korevaar-Schoen能量与插值性质，证明满足弱接触方程的映射属于特定水平分数阶Sobolev空间，建立插值定理并刻画Korevaar-Schoen能量的极小化子。
中文摘要 AI 辅助
本文研究Carnot群之间的分数阶Sobolev映射理论，重点关注Korevaar-Schoen能量泛函与插值性质。我们证明Carnot群上的分数阶Sobolev映射满足弱接触方程，将欧氏框架下的经典结果推广到次黎曼框架。主要结果包括：(1) 对任意足够小的$\lambda>0$，满足弱接触方程的分数阶Sobolev映射属于水平分数阶Sobolev空间$W_H^{s-\lambda,q}(\Omega;G_2)$；(2) Carnot群之间分数阶Sobolev映射的插值定理，证明当$s \to 1$取极限时可恢复一阶Sobolev空间；(3) 严格处理Carnot群上的Korevaar-Schoen能量，包括通过水平微分形式刻画其极小化子。
英文摘要
This paper investigates the theory of fractional Sobolev mappings between Carnot groups, with a particular focus on the Korevaar-Schoen energy functional and interpolation properties. We establish that weak contact equations hold for fractional Sobolev mappings on Carnot groups, generalizing classical results from the Euclidean setting to the sub-Riemannian framework. Our main results include: (1) the proof that fractional Sobolev mappings satisfying weak contact equations belong to the horizontal fractional Sobolev space $W_H^{s-λ,q}(Ω;G_2)$ for any sufficiently small $λ>0$; (2) interpolation theorems for fractional Sobolev mappings between Carnot groups, demonstrating that the limiting behavior as $s \to 1$ recovers the first-order Sobolev space; (3) a rigorous treatment of the Korevaar-Schoen energy on Carnot groups, including a characterization of its minimizers via horizontal differential forms.
