arXiv subjects

Luc Pétiard

Publications and source records attributed to Luc Pétiard.

2 recordsLinked to original sources

arXiv2019

Eigenvalues of the Laplacian with density

Let $(M,g)$ be a compact Riemannian manifold with a boundary of class $\mathscr{C}^{1}$. We are interested in the spectrum of the weighted Laplacian on $M$ with Neumann boundary conditions. More precisely, given $ρ$ and $σ$ two positive functions on $M$, we study the eigenvalues of the equation $-\operatorname{div}(σ\nabla u)=λρu$. Inspired by a recent work of B. Colbois and A. El Soufi, we investigate upper bounds for the eigenvalues in the case where $σ=ρ^α$, $α>0$. We show that $α= \frac{n-2}{n}$ plays a critical role in the estimation of the spectrum when the total mass of $ρ$ is fixed.

math.SP↗

arXiv2017

Counter-example to a Kröger type spectral inequality

Given a Riemannian manifold, Weyl's law indicates how the spectrum of the Laplacian behaves asymptotically. Because of that result, there has been a growing interest in finding geometrical bounds compatible with this law. In the case of hypersurfaces, the isoperimetric ratio is a natural geometrical quantity, that allows to bound the spectrum from above. We investigate the problem and find an example of hypersurface where the eigenvalues are minorated by the isoperimetric ratio.

math.SP↗