Research Topics

In the field of shape optimization, I am interested in minimal partitions for eigenvalues. In collaboration with V. Bonnaillie-Noël, I have looked for minimal partitions of this type using asymptotic and numerical methods.

The topic of minimal partitions led me to the study of nodal sets for Laplacian eigenfunctions, in various domains and with various boundary conditions. In particular, I have worked on generalizations of Pleijel's nodal domain theorem.

I am also studying the Schrödinger operators associated with a singular magnetic potential of Aharonov-Bohm type. With L. Abatangelo, V. Felli and L. Hillairet, we are investigating the way the eigenvalues depend on the position of the singularities.

From 2010 to 2016, I have participated in the activities of the research groups GAOS and OPTIFORM, on shape optimization, funded by the ANR (Agence Nationale pour la Recherche).

Publications

• L. Abatangelo, V. Felli et C. Léna, On Aharonov-Bohm operators with two colliding poles. Adv. Nonlinear Stud. 17, 2 pp. 283–296 (2017). doi: 10.1515/ans-2017-0004 Preprint
• V. Bonnaillie-Noël et C. Léna, Spectral minimal partitions for a familly of tori. Exp. Math. 26, 4 pp. 381–395 (2017). doi: 10.1080/10586458.2016.1183154 Preprint
• C. Léna, Courant-sharp eigenvalues of the three-dimensional square torus. Proc. Amer. Math. Soc., 144, 9 pp. 3949-3958 (2016). doi: 10.1090/proc/13148 Preprint
• C. Léna, Courant-sharp eigenvalues of a two-dimensional torus. C. R. Math. Acad. Sci. Paris, 353, 6 pp. 535–539 (2015). doi: 10.1016/j.crma.2015.03.014 Preprint
• C. Léna, Eigenvalues variations for Aharonov-Bohm operators. J. Math. Phys., 56, 1 (2015). doi: 10.1063/1.4905647 Preprint
• V. Bonnaillie-Noël and C. Léna, Spectral minimal partitions of a sector. Discrete Contin. Dyn. Syst. Ser. B, 19, 1 pp. 27-53 (2014). doi: 10.3934/dcdsb.2014.19.27 Preprint

Preprints

• C. Léna, Examples of spectral minimal partitions. 8 p (2017). arXiv:1701.05058 [math.AP] PDF
• C. Léna, Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions. 13 p (2016). arXiv:1609.02331 [math.AP] PDF
• L. Abatangelo, V. Felli, L. Hillairet and C. Léna, Spectral stability under removal of small capacity sets and applications to Aharonov-Bohm operators. 31 p (2016). arXiv:1611.06750 [math.AP] PDF
• C. Léna, On the parity of the number of nodal domains for an eigenfunction of the Laplacian on tori. 5 p (2015). arXiv:1504.03944 [math.AP] PDF

Selected talks

• Generalizations of Pleijel's nodal domain theorem. Workshop - Geometric Spectral Theory, Neuchâtel, 19 June 2017. PDF
• Generalizations of Pleijel's nodal domain theorem. Seminar of the Group of Mathematical Physics, Lisbon, 4 April 2017. PDF
• Spectral estimates for punctured domains and Aharonov-Bohm operators. Seminar of Spectral Theory and Geometry, Neuchâtel, 22 March 2017. Part 1Part 2
• Review of Spectral Minimal Partitions. Meeting of the ZIF cooperation group Discrete and Continuous Models in the Theory of Networks, Bielefeld, 7 March 2017. PDF
• Examples of spectral minimal partitions. Conference - Bruxelles-Torino talks in PDE's, Turin, 5 May 2016. PDF
• Nodal patterns for the Laplacian on flat tori. Conference - Variational Perspectives, Turin (Politecnico di Torino), 7 March 2016. PDF
• Nodal patterns of the Laplacian on thin domains. Conference - Asymptotic Analysis and Spectral Theory, Orsay, 6 October 2015. PDF
• The Courant-sharp propery for flat tori. Conference - Calculus of variations and PDEs, Chambéry, 25 September 2015. PDF
• On the number of nodal domains for flat tori. SMS - Geometric and Computational Spectral Theory, Montréal (UQÀM), 24 June 2015. PDF
• Eigenvalues variations for Aharonov-Bohm operators. Workshop - Magnetic fields and semi-classical analysis, Rennes, 22 May 2015. PDF
• Minimal partitions of flat tori. Winter school on spectral theory and shape optimization problems for elliptic PDEs, Milan (Milano-Biccoca), 12 February 2015. PDF
• Partitions minimales d'un tore plat. Workshop Calculus of Variations, Paris (Paris-Dauphine University), 13 October 2014. PDF
• Valeurs propres des opérateurs d'Aharonov-Bohm. Séminaire EDP et Physique Mathématique, Bordeaux, 19 March 2014. PDF
• Eigenvalues variations for Aharonov-Bohm operators. Annual meeting of GDR DynQua, Roscoff, 5 February 2014. PDF
• Examples of nodal minimal partitions. Follow-up workshop on Selected Topics in Spectral Theory, Vienna, 27 January 2011