Rola Kiwan | Searcharxiv

arXiv2021

Courant-sharp eigenvalues of compact flat surfaces: Klein bottles and cylinders

The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, Möbius strips,\ldots . A natural toy model for further investigations is the flat Klein bottle, a non-orientable surface with Euler characteristic $0$, and particularly the Klein bottle associated with the square torus, whose eigenvalues have higher multiplicities. In this note, we prove that the only Courant-sharp eigenvalues of the flat Klein bottle associated with the square torus (resp. with square fundamental domain) are the first and second eigenvalues. We also consider the flat cylinders $(0,π) \times \mathbb{S}^1_r$ where $r \in \{0.5,1\}$ is the radius of the circle $\mathbb{S}^1_r$, and we show that the only Courant-sharp Dirichlet eigenvalues of these cylinders are the first and second eigenvalues.

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arXiv2020

Courant-sharp property for Dirichlet eigenfunctions on the Möbius strip

The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, \ldots . A natural toy model for further investigations is the Möbius strip, a non-orientable surface with Euler characteristic $0$, and particularly the "square" Möbius strip whose eigenvalues have higher multiplicities. In this case, we prove that the only Courant-sharp Dirichlet eigenvalues are the first and the second, and we exhibit peculiar nodal patterns.

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arXiv2015

Dirichlet eigenfunctions on the cube, sharpening the Courant nodal inequality

This paper is devoted to the refine analysis of Courant's theorem for the Dirichlet Laplacian. Many papers (and some of them quite recent) have investigated in which cases this inequality in Courant's theorem is an equality: Pleijel, Helffer--Hoffmann-Ostenhof--Terracini, Helffer--Hoffmann-Ostenhof, Bérard-Helffer, Helffer--Persson-Sundqvist, Léna, Leydold. All these results were devoted to $(2D)$-cases in open sets in $\mathbb R^2$ or in surfaces like $\mathbb S^2$ or $\mathbb T^2$. The aim of the current paper is to look for analogous results for domains in $\mathbb{R}^3$ and, as $Å.$Pleijel was suggesting in his 1956 founding paper, for the simplest case of the cube. More precisely, we will prove that the only eigenvalues of the Dirichlet Laplacian which are Courant sharp are the two first eigenvalues.

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arXiv2007

Where to place a spherical obstacle so as to maximize the second Dirichlet eigenvalue

We prove that among all doubly connected domains of $\mathbb{R}^n$ bounded by two spheres of given radii, the second eigenvalue of the Dirichlet Laplacian achieves its maximum when the spheres are concentric (spherical shell). The corresponding result for the first eigenvalue has been established by Hersch in dimension 2, and by Harrell, Kröger and Kurata and Kesavan in any dimension. We also prove that the same result remains valid when the ambient space $\mathbb{R}^n$ is replaced by the standard sphere $\mathbb{S}^n$ or the hyperbolic space $\mathbb{H}^n$ .

math.MG↗

arXiv2007

Extremal first Dirichlet eigenvalue of doubly connected plane domains and dihedral symmetry

We deal with the following eigenvalue optimization problem: Given a bounded domain $D\subset \R^2$, how to place an obstacle $B$ of fixed shape within $D$ so as to maximize or minimize the fundamental eigenvalue $λ_1$ of the Dirichlet Laplacian on $D\setminus B$. This means that we want to extremize the function $ρ\mapsto λ_1(D\setminus ρ(B))$, where $ρ$ runs over the set of rigid motions such that $ρ(B)\subset D$. We answer this problem in the case where both $D$ and $B$ are invariant under the action of a dihedral group $\mathbb{D}_n$, $n\ge2$, and where the distance from the origin to the boundary is monotonous as a function of the argument between two axes of symmetry. The extremal configurations correspond to the cases where the axes of symmetry of $B$ coincide with those of $D$.

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