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Girouard, Alexandre

Nom
Girouard, Alexandre
Affiliation principale
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https://libra.unine.ch/handle/123456789/464

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  • 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
    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.