27 citations · 95 across the 24 of their papers we have counts for
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Equivariant Steklov homogenisation and stability of spectral representations
Samuel Audet-Beaumont, Maxime Fortier Bourque, Alexandre Girouard
We prove that isolated spectral clusters preserve their finite-group representation under equivariant spectral convergence. In particular, an irreducible limiting eigenspace cannot…
An elementary counterexample to Escobar's Steklov conjecture on the three-ball
Alexandre Girouard, Thomas Hélière
We give an explicit polynomial deformation of the Euclidean metric on the unit three-ball which has positive Ricci curvature and strictly convex boundary but violates Escobar's con…
Upper bounds for the Steklov eigenvalues of warped products
Jade Brisson, Bruno Colbois, Alexandre Girouard +1
We obtain upper bounds for the Steklov eigenvalues of warped products , where is a compact Riemannian manifold with boundary and is a closed Riemannian manifold…
The exterior Steklov problem for Euclidean domains
Lukas Bundrock, Alexandre Girouard, Denis S. Grebenkov +2
We investigate the Steklov eigenvalue problem in an exterior Euclidean domain. First, we present several formulations of this problem and establish the equivalences between them. N…
Upper bounds for the second nonzero eigenvalue of the Laplacian via folding and conformal volume
Mehdi Eddaoudi, Alexandre Girouard
We prove an upper bound for the volume-normalized second nonzero eigenvalue of the Laplace operator on closed Riemannian manifold, in terms of the conformal volume. This bound prov…
Some recent developments on the Steklov eigenvalue problem
Bruno Colbois, Alexandre Girouard, Carolyn Gordon +1
The Steklov eigenvalue problem, first introduced over 125 years ago, has seen a surge of interest in the past few decades. This article is a tour of some of the recent developments…
Metric upper bounds for Steklov and Laplace eigenvalues
Bruno Colbois, Alexandre Girouard
We prove two upper bounds for the Steklov eigenvalues of a compact Riemannian manifold with boundary. The first involves the volume of the manifold and of its boundary, as well as…
The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander's rediscovered manuscript
Alexandre Girouard, Mikhail Karpukhin, Michael Levitin +1
How close is the Dirichlet-to-Neumann (DtN) map to the square root of the corresponding boundary Laplacian? This question has been actively investigated in recent years. Somewhat s…
Large Steklov eigenvalues via homogenisation on manifolds
Alexandre Girouard, Jean Lagacé
Using methods in the spirit of deterministic homogenisation theory we obtain convergence of the Steklov eigenvalues of a sequence of domains in a Riemannian manifold to weighted La…
Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems
Alexandre Girouard, Mikhail Karpukhin, Jean Lagacé
We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and ot…
Robin spectrum: two disks maximize the third eigenvalue
Alexandre Girouard, Richard S. Laugesen
The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the third eigenvalue of a disjoint union of two disks, provided th…
Hypersurfaces of Euclidean space with prescribed boundary and small Steklov eigenvalues
Bruno Colbois, Alexandre Girouard, Antoine Métras
Given a smooth compact hypersurface with boundary , we prove the existence of a sequence of hypersurfaces with the same boundary as , such that each Stek…
The Steklov and Laplacian spectra of Riemannian manifolds with boundary
Bruno Colbois, Alexandre Girouard, Asma Hassannezhad
Given two compact Riemannian manifolds with boundary and such that their respective boundaries and admit neighborhoods and which are isometric,…
The Steklov spectrum of cuboids
Alexandre Girouard, Jean Lagacé, Iosif Polterovich +1
The paper is concerned with the Steklov eigenvalue problem on cuboids of arbitrary dimension. We prove a two-term asymptotic formula for the counting function of Steklov eigenvalue…
Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space
Bruno Colbois, Alexandre Girouard, Katie Gittins
We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on th…
Compact manifolds with fixed boundary and large Steklov eigenvalues
Bruno Colbois, Ahmad El Soufi, Alexandre Girouard
Let M be a compact Riemannian manifold with boundary. Let b>0 be the number of connected components of its boundary. For manifolds of dimension at least 3, we prove that it is poss…
Steklov eigenvalues and quasiconformal maps of simply connected planar domains
A. Girouard, R. S. Laugesen, B. A. Siudeja
We investigate isoperimetric upper bounds for sums of consecutive Steklov eigenvalues of planar domains. The normalization involves the perimeter and scale-invariant geometric fact…
Spectral geometry of the Steklov problem
Alexandre Girouard, Iosif Polterovich
The Steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichl…
The Steklov spectrum of surfaces: asymptotics and invariants
Alexandre Girouard, Leonid Parnovski, Iosif Polterovich +1
We obtain precise asymptotics for the Steklov eigenvalues on a compact Riemannian surface with boundary. It is shown that the number of connected components of the boundary, as wel…
The spectral gap of graphs and Steklov eigenvalues on surfaces
Bruno Colbois, Alexandre Girouard
Using expander graphs, we construct a sequence of smooth compact surfaces with boundary of perimeter N, and with the first non-zero Steklov eigenvalue uniformly bounded away from z…
Uniform stability of the Dirichlet spectrum for rough outer perturbations
Bruno Colbois, Alexandre Girouard, Mette Iversen
The goal of this paper is to study the Dirichlet eigenvalues of bounded domains . With a local spectral stability requirement on , we show that the difference of th…
Upper bounds for Steklov eigenvalues on surfaces
Alexandre Girouard, Iosif Polterovich
We give explicit isoperimetric upper bounds for all Steklov eigenvalues of a compact orientable surface with boundary, in terms of the genus, the length of the boundary, and the nu…
Isoperimetric control of the Steklov spectrum
Bruno Colbois, Ahmad El Soufi, Alexandre Girouard
Let N be a complete Riemannian manifold of dimension n+1 whose Riemannian metric g is conformally equivalent to a metric with non-negative Ricci curvature. The normalized Steklov e…
Shape optimization for low Neumann and Steklov eigenvalues
Alexandre Girouard, Iosif Polterovich
We give an overview of results on shape optimization for low eigenvalues of the Laplacian on bounded planar domains with Neumann and Steklov boundary conditions. These results shar…