论文标题
边界上的Riemannian歧管上的外部差分形式的独特定理
A unique continuation theorem for exterior differential forms on Riemannian manifolds with boundary
论文作者
论文摘要
Aronszajn，Krzywicki和Szarski在\ cite {aks62}中证明了差分形式的强烈独特延续结果，满足了带有空边界的Riemannian歧管上的一阶差分不平等。本文假设在差分形式上有适当的边界条件，将此结果扩展到了带有非空边界的Riemannian歧管的设置。然后，我们提出了该扩展结果的一些应用。也就是说，我们表明，零谐波和迪里奇特表格的零组的豪斯多夫尺寸以及卷曲操作员的特征菲尔德（以$ 3 $ -Manifolds为单位），具有至少$ 2 $。同样，这些界限是在没有边界的歧管的环境中知道的，因此优点再次是边界点的包含。
Aronszajn, Krzywicki and Szarski proved in \cite{AKS62} a strong unique continuation result for differential forms, satisfying a certain first order differential inequality, on Riemannian manifolds with empty boundary. The present paper extends this result to the setting of Riemannian manifold with non-empty boundary, assuming suitable boundary conditions on the differential forms. We then present some applications of this extended result. Namely, we show that the Hausdorff dimension of the zero set of harmonic Neumann and Dirichlet forms, as well as eigenfields of the curl operator (on $3$-manifolds), has codimension at least $2$. Again, these bounds were known in the setting of manifolds without boundary, so that the merit is once more the inclusion of boundary points.
