Corentin Léna

Corentin Léna
Université de Neuchâtel | UniNE · Institut de mathématiques (IMA)

PhD in Mathematics

About

28
Publications
1,377
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137
Citations
Introduction
I work in mathematical analysis an numerical analysis. My main research topics are the eigenvalues and eigenfunctions of elliptic operators, and shape optimization, especially minimal partition problems. I am currently doing a postdoc in the Department of Mathematics of Stockholm University, under the supervision of Pavel Kurasov.
Additional affiliations
September 2018 - present
Stockholm University
Position
  • PostDoc Position
Description
  • Postdoc Position in the Research group in Analysis, under the supervision of Pavel Kurasov.
July 2017 - July 2018
University of Lisbon
Position
  • PostDoc Position
Description
  • Postdoc Position in the project "Shape optimization in high dimensions"; Principal Investigator: Pedro Antunes.
February 2015 - June 2017
Università degli Studi di Torino
Position
  • PostDoc Position
Description
  • Junior member of the COMPAT Project (ERC-2013-AG); Principal Investigator: Susanna Terracini.
Education
September 2010 - December 2013
Université Paris-Sud 11
Field of study
  • Mathematics
September 2008 - August 2010
Ecole normale supérieure de Cachan
Field of study
  • Mathematics
September 2007 - September 2010
Université Paris-Sud 11
Field of study
  • Mathematics

Publications

Publications (28)
Preprint
Taking advantage from the so-called "Lemma on small eigenvalues" by Colin de Verdi\`{e}re, we study ramification for multiple eigenvalues of the Dirichlet Laplacian in bounded perforated domains. The asymptotic behavior of multiple eigenvalues turns out to depend on the asymptotic expansion of suitable associated eigenfunctions. We treat the case o...
Preprint
We are concerned in this paper with the real eigenfunctions of Schr\"odinger operators. We prove an asymptotic upper bound for the number of their nodal domains, which implies in particular that the inequality stated in Courant's theorem is strict, except for finitely many eigenvalues. Results of this type originated in 1956 with Pleijel's Theorem...
Article
Full-text available
We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partition...
Article
In this paper we study the asymptotic behavior of $u$-capacities of small sets and its application to the analysis of the eigenvalues of the Dirichlet-Laplacian on a bounded planar domain with a small hole. More precisely, we consider two (sufficiently regular) bounded open connected sets $\Omega$ and $\omega$ of $\mathbb{R}^2$, containing the orig...
Preprint
Full-text available
We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partition...
Preprint
Trace formulas play a central role in the study of spectral geometry and in particular of quantum graphs. The basis of our work is the result by Kurasov which links the Euler characteristic $\chi$ of metric graphs to the spectrum of their standard Laplacian. These ideas were shown to be applicable even in an experimental context where only a finite...
Article
We revisit an interesting example proposed by Maria Hoffmann-Ostenhof, the second author and Nikolai Nadirashvili of a bounded domain in R 2 for which the second eigenvalue of the Dirichlet Laplacian has multiplicity 3. We also analyze carefully the first eigenvalues of the Laplacian in the case of the disk with two symmetric cracks placed on a sma...
Article
We deal with the sharp asymptotic behaviour of eigenvalues of elliptic operators with varying mixed Dirichlet–Neumann boundary conditions. In case of simple eigenvalues, we compute explicitly the constant appearing in front of the expansion’s leading term. This allows inferring some remarkable consequences for Aharonov–Bohm eigenvalues when the sin...
Preprint
In this paper we study the asymptotic behavior of $u$-capacities of small sets and its application to the analysis of the eigenvalues of the Dirichlet-Laplacian on a bounded planar domain with a small hole. More precisely, we consider two (sufficiently regular) bounded open connected sets $\Omega$ and $\omega$ of $\mathbb{R}^2$, containing the orig...
Preprint
Full-text available
We revisit an interesting example proposed by Maria Hoffmann-Ostenhof, the second author and Nikolai Nadirashvili of a bounded domain in R2 for which the second eigenvalue of the Dirichlet Laplacian has multiplicity three. We also analyze carefully the first eigenvalues of the Laplacian in the case of the disk with two symmetric cracks placed on a...
Preprint
Full-text available
We obtain upper bounds for the Courant-sharp Neumann and Robin eigenvalues of an open, bounded, connected set in $\mathbb{R}^n$ with $C^2$ boundary. In the case where the set is also assumed to be convex, we obtain explicit upper bounds in terms of some of the geometric quantities of the set.
Preprint
Full-text available
The present paper deals with the asymptotic behaviour of eigenvalues of elliptic operators with varying mixed Dirichlet-Neumann boundary conditions. More precisely we establish a sharp relation between the order of vanishing of a Dirichlet eigenfunction at a boundary point and the asymptotics of the eigenvalue variation under homogeneous boundary c...
Article
Full-text available
We study a minimal partition problem on the flat rectangular torus. We give a partial review of the existing literature, and present some numerical and theoretical work recently published elsewhere by V. Bonnaillie-No{\"e}l and the author, with some improvements.
Article
Full-text available
In this paper, we show that equality in Courant's nodal domain theorem can only be reached for a finite number of eigenvalues of the Neumann Laplacian, in the case of an open, bounded and connected set in R n with a C 1,1 boundary. This result is analogous to Pleijel's nodal domain theorem for the Dirichlet Laplacian (1956). It confirms, in all dim...
Article
We consider Aharonov-Bohm operators with two poles and prove sharp asymptotics for simple eigenvalues as the poles collapse at an interior point out of nodal lines of the limit eigenfunction.
Article
Full-text available
We first establish a sharp relation between the order of vanishing of a Dirichlet eigenfunction at a point and the leading term of the asymptotic expansion of the Dirichlet eigenvalue variation, as a removed compact set concentrates at that point. Then we apply this spectral stability result to the study of the asymptotic behaviour of eigenvalues o...
Research
We study partitions of the two-dimensional flat torus of legnth 1 and width b into k domains, with k a real parameter in (0,1] and k an integer. We look for partitions which minimize the energy, defined as the largest first eigenvalue of the Dirichlet Laplacian on the domains of the partition. We are in particular interested in the way these minima...
Article
Full-text available
We study partitions of the two-dimensional flat torus into k domains, with b a real parameter in (0, 1] and k an integer. We look for partitions which minimize the energy, defined as the largest first eigenvalue of the Dirichlet Laplacian on the domains of the partition. We are in particular interested in the way these minimal partitions change whe...
Article
Full-text available
In this paper, we determine, in the case of the Laplacian on the flat three-dimensional torus $(\mathbb{R}/\mathbb{Z})^3$, all the eigenvalues having an eigenfunction which satisfies the Courant nodal domains theorem with equality (Courant-sharp situation). Following the strategy of {\AA}. Pleijel (1956), the proof is a combination of an explicit l...
Article
Full-text available
We study how the eigenvalues of a magnetic Schrodinger operator of Aharonov-Bohm type depend on the singularities of its magnetic potential. We consider a magnetic potential defined everywhere in R-2 except at a finite number of singularities, so that the associated magnetic field is zero. On a fixed planar domain, we define the corresponding magne...
Article
In this paper, we determine, in the case of the Laplacian on the flat two-dimensional torus (R/Z) 2 , all the eigenvalues having an eigenfunction which satisfies Courant's theorem with equality (Courant-sharp situation). Following the strategy o A. Pleijel (1956), the proof is a combination of a lower bound a la Weyl) of the counting function, with...
Article
Full-text available
In this article, we are interested in determining spectral minimal k-partitions for angular sectors. We first deal with the nodal cases for which we can determine explicitly the minimal partitions. Then, in the case where the minimal partitions are not nodal domains of eigenfunctions of the Dirichlet Laplacian, we analyze the possible topologies of...
Thesis
Full-text available
This work is concerned with the problem of minimal partitions, at the interface between spectral theory and shape optimization. A general introduction gives a precise statement of the problem and recall results, mainly due to B. Helffer, T. Hoffmann-Ostenhof and S.Terracini, that are used in the rest of the thesis.The first chapter is an asymptotic...