On Courant and Pleijel theorems for sub-Riemannian Laplacians
arxiv(2024)
摘要
We are interested in the number of nodal domains of eigenfunctions of
sub-Laplacians on sub-Riemannian manifolds. Specifically, we investigate the
validity of Pleijel's theorem, which states that, as soon as the dimension is
strictly larger than 1, the number of nodal domains of an eigenfunction
corresponding to the k-th eigenvalue is strictly (and uniformly, in a certain
sense) smaller than k for large k.
In the first part of this paper we reduce this question from the case of
general sub-Riemannian manifolds to that of nilpotent groups.
In the second part, we analyze in detail the case where the nilpotent group
is a Heisenberg group times a Euclidean space. Along the way we improve known
bounds on the optimal constants in the Faber-Krahn and isoperimetric
inequalities on these groups.
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