arXiv subjects

Corentin Léna

Publications and source records attributed to Corentin Léna.

23 records · Page 2Linked to original sources

arXiv2016

Pleijel's nodal domain theorem for Neumann and Robin eigenfunctions

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 dimensions, a conjecture formulated by Pleijel, which had already been solved by I. Polterovich for a two-dimensional domain with a piecewise-analytic boundary (2009). We also show that the argument and the result extend to a class of Robin boundary conditions.

math.AP↗

arXiv2016

Spectral minimal partitions for a family of tori

We study partitions of the rectangular two-dimensional flat torus of length 1 and width b into k domains, with b a parameter in (0, 1] and k an integer. We look for partitions which minimize the energy, definedas the largest first eigenvalue of the Dirichlet Laplacian on the domains of the partition. We are inparticular interested in the way these minimal partitions change when b is varied. We present herean improvement, when k is odd, of the results on transition values of b established by B. Helffer andT. Hoffmann-Ostenhof (2014) and state a conjecture on those transition values. We establishan improved upper bound of the minimal energy by explicitly constructing hexagonal tilings of thetorus. These tilings are close to the partitions obtained from a systematic numerical study based on an optimization algorithm adapted from B. Bourdin, D. Bucur, and {É}. Oudet (2009). These numerical results also support our conjecture concerning the transition values and give betterestimates near those transition values.

math.AP↗

arXiv2015

Courant-sharp eigenvalues of the three-dimensional square torus

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 Å. Pleijel (1956), the proof is a combination of an explicit lower bound of the counting function and a Faber-Krahn-type inequality for domains on the torus, deduced as, in the work of P. Bérard and D. Meyer (1982), from an isoperimetric inequality. This inequality relies on the work of L. Hauswirth, J. Perez, P. Romon, and A. Ros (2004) on the periodic isoperimetric problem.

math.SP↗

arXiv2015

Courant-sharp eigenvalues of a two-dimensional torus

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 an explicit remainder term, and of a Faber--Krahn inequality for domains on the torus (deduced as in B{é}rard-Meyer from an isoperimetric inequality), with an explicit upper bound on the area.

math.AP↗

arXiv2015

On the parity of the number of nodal domains for an eigenfunction of the Laplacian on tori

In this note, we discuss a question posed by T. Hoffmann-Ostenhof concerning the parity of the number of nodal domains for a non-constant eigenfunction of the Laplacian on flat tori. We present two results. We first show that on the torus $(\mathbb{R}/2π\mathbb{Z})^{2}$, a non-constant eigenfunction has an even number of nodal domains. We then consider the torus $(\mathbb{R}/2π\mathbb{Z})\times(\mathbb{R}/2ρπ\mathbb{Z})\,$, with $ρ=\frac{1}{\sqrt{3}}\,$, and construct on it an eigenfunction with three nodal domains.

math.AP↗