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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.
  • Publication
    Métadonnées seulement
  • Publication
    Métadonnées seulement
    Isoperimetric control of the Steklov spectrum
    (2011-6-21)
    Colbois, Bruno 
    ;
    El Soufi, Ahmad
    ;
    Girouard, Alexandre 
    We prove that the normalized Steklov eigenvalues of a bounded domain in a complete Riemannian manifold are bounded above in terms of the inverse of the isoperimetric ratio of the domain. Consequently, the normalized Steklov eigenvalues of a bounded domain in Euclidean space, hyperbolic space or a standard hemisphere are uniforml bounded above. On a compact surface with boundary, we obtain uniform bounds for the normalized Steklov eigenvalues in terms of the genus. We also establish a relationship between the Steklov eigenvalues of a domain and the eigenvalues of the Laplace-Beltrami operator on its boundary hypersurface.
  • Publication
    Métadonnées seulement
    Bounding the eigenvalues of the Laplace-Beltrami operator on compact submanifolds
    (2010-1-21)
    Colbois, Bruno 
    ;
    Dryden, Emily B
    ;
    El Soufi, Ahmad
    We give upper bounds for the eigenvalues of the La-place-Beltrami operator of a compact m-dimensional submanifold M of R^{m+p}. Besides the dimension and the volume of the submanifold and the order of the eigenvalue, these bounds depend on either the maximal number of intersection points of M with a p-plane in a generic position (transverse to M), or an invariant which measures the concentration of the volume of M in R^{m+p}. These bounds are asymptotically optimal in the sense of the Weyl law. On the other hand, we show that even for hypersurfaces (i.e., when p=1), the first positive eigenvalue cannot be controlled only in terms of the volume, the dimension and (for m>2) the differential structure.
  • Publication
    Métadonnées seulement
    Extremal g-invariant eigenvalues of the Laplacian of g-invariant metrics
    (2008-12-21)
    Colbois, Bruno 
    ;
    Dryden, Emily B
    ;
    El Soufi, Ahmad
    The study of extremal properties of the spectrum often involves restricting the metrics under consideration. Motivated by the work of Abreu and Freitas in the case of the sphere S-2 endowed with S-1-invariant metrics, we consider the subsequence lambda(G)(k) of the spectrum of a Riemannian manifold M which corresponds to metrics and functions invariant under the action of a compact Lie group G. If. G has dimension at least 1, we show that the functional lambda(G)(k) admits no extremal metric under volume-preserving G-invariant deforma- tions. If, moreover, M has dimension at least three, then the functional lambda(G)(k) is unbounded when restricted to any conformal class of G-invariant metrics of fixed volume. As a special case of this, we can consider the standard 0(n)-action on S-n; however, if we also require the metric to be induced by an embedding of S-n in Rn+1, we get an optimal upper bound on lambda(G)(k).
  • Publication
    Métadonnées seulement
    Eigenvalues of the laplacian acting on p-forms and metric conformal deformations
    (2006)
    Colbois, Bruno 
    ;
    El Soufi, Ahmad
    Let (M, g) be a compact connected orientable Riemannian manifold of dimension n >= 4 and let lambda(k,p)(g) be the k-th positive eigenvalue of the Laplacian. Delta g,p = dd* + d* d acting on differential forms of degree p on M. We prove that the metric g can be conformally deformed to a metric g', having the same volume as g, with arbitrarily large lambda 1, p(g') for all p is an element of [2,n-2]. Note that for the other values of p, that is p = 0, 1, n-1 and n, one can deduce from the literature that, for all k > 0, the k-th eigenvalue lambda(k,p) is uniformly bounded on any conformal class of metrics of fixed volume on M. For p = 1, we show that, for any positive integer N, there exists a metric g(N) conformal to g such that, for all k