On Courant and Pleijel theorems for sub-Riemannian Laplacians
Seminari di Analisi Matematica dell'Università e del Politecnico di Torino
Date: Tuesday, 19 November 2024 13:00 - 14:00
Venue: Bernard Helffer - Aula Buzano (DISMA Politecnico di Torino)
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 one, 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 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.
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 one, 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 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.
Speaker
Bernard Helffer
Affiliation
Université Paris-Sud
Room
Aula Buzano
Proposed by
Patrick Wahlberg
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