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Colbois, Bruno

Nom
Colbois, Bruno
Affiliation principale
Site web
Fonction
Professeur ordinaire
Email
Bruno.Colbois@unine.ch
Identifiants
https://libra.unine.ch/handle/123456789/456
_
0000-0002-8514-4315

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  • Publication
    Accès libre
    Eigenvalues upper bounds for the magnetic Schrödinger operator
    (2022-3-6)
    Colbois, Bruno 
    ;
    El Soufi, Ahmad
    ;
    Ilias, Saïd
    ;
    Savo, Alessandro
  • Publication
    Accès libre
    Compact manifolds with fixed boundary and large Steklov eigenvalues
    (2019-8-22)
    Colbois, Bruno 
    ;
    El Soufi, Ahmad
    ;
    Girouard, Alexandre 
    Let $(M,g)$ be a compact Riemannian manifold with boundary. Let $b>0$ be the number of connected components of its boundary. For manifolds of dimension $\geq 3$, we prove that for $j=b+1$ it is possible to obtain an arbitrarily large Steklov eigenvalue $\sigma_j(M,e^\delta g)$ using a conformal perturbation $\delta\in C^\infty(M)$ which is supported in a thin neighbourhood of the boundary, with $\delta=0$ on the boundary. For $j\leq b$, it is also possible to obtain arbitrarily large eigenvalues, but the conformal factor must spread throughout the interior of $M$. This is in stark contrast with the situation for the eigenvalues of the Laplace operator, for which the supremum is bounded in each fixed conformal class. In fact, when working in a fixed conformal class, it is known that the volume of $(M,e^\delta g)$ has to tend to infinity in order for some $\sigma_j$ to become arbitrarily large. We also prove that it is possible to obtain large eigenvalues while keeping different boundary components arbitrarily close to each others, by constructing a convenient Riemannian submersion.
  • Publication
    Accès libre
    Spectrum of the Laplacian with weights
    (2019-3-4)
    Colbois, Bruno 
    ;
    El Soufi, Ahmad
    Given a compact Riemannian manifold $(M,g)$ and two positive functions $\rho$ and $\sigma$, we are interested in the eigenvalues of the Dirichlet energy functional weighted by $\sigma$, with respect to the $L^2$ inner product weighted by $\rho$. Under some regularity conditions on $\rho$ and $\sigma$, these eigenvalues are those of the operator $-\rho^{-1} \mbox{div}(\sigma \nabla u)$ with Neumann conditions on the boundary if $\partial M\ne \emptyset$. We investigate the effect of the weights on eigenvalues and discuss the existence of lower and upper bounds under the condition that the total mass is preserved.
  • Publication
    Accès libre
    Eigenvalues of the Laplacian on a compact manifold with density
    (2015-3-1)
    Colbois, Bruno 
    ;
    El Soufi, Ahmad
    ;
    Savo, Alessandro
  • Publication
    Accès libre
    Extremal eigenvalues of the Laplacian on Euclidean domains
    (2014-10-1)
    Colbois, Bruno 
    ;
    El Soufi, Ahmad
    We investigate properties of the sequences of extremal values that could be achieved by the eigenvalues of the Laplacian on Euclidean domains of unit volume, under Dirichlet and Neumann boundary conditions, respectively. In a second part, we study sequences of extremal eigenvalues of the Laplace-Beltrami operator on closed surfaces of unit area.