Courant-sharp eigenvalues of the three-dimensional square torus (Q2814414)

Consider the eigenvalues \(\lambda_k \geq 0\), \(k \geq 1\), of the Laplacian on a flat torus \(\mathbb{T}^3 = \mathbb{R}^3 / \mathbb{Z}^3\), ordered by increasing size and repeated according to their multiplicities. According to a classical theorem of Courant on domains, but readily adaptable to \(\mathbb{T}^3\), the zero set of any eigenfunction associated with \(\lambda_k\) can divide \(\mathbb{T}^3\) into at most \(k\) connected components, the nodal domains. A refinement due to Pleijel shows that the number of \(k \geq 1\) for which equality holds, i.e., for which such an eigenfunction has exactly \(k\) nodal domains, is finite (we call \(\lambda_k\) Courant-sharp in this case).NEWLINENEWLINEIn the last couple of years there has been considerable activity to determine the set of all such Courant-sharp eigenvalues of the Laplacian on various concrete domains such as squares, cubes, disks and some triangles, equipped with Dirichlet or Neumann boundary conditions. In the present paper the case \(\mathbb{T}^3\) is treated.NEWLINENEWLINEThe main result states that the Courant-sharp eigenvalues correspond exactly to \(k=1,\ldots,7\) (noting that \(\lambda_1=0\) and \(\lambda_2\) has multiplicity \(6\)).NEWLINENEWLINEThe proof relies on a new inequality of isoperimetric type for small enough open sets \(\Omega \subset \mathbb{T}^3\) based on work of \textit{L. Hauswirth} et al. [Trans. Am. Math. Soc. 356, No. 5, 2025--2047 (2004; Zbl 1046.52002)], from which an estimate on the first Dirichlet Laplacian eigenvalue on such sets follows: denoting this eigenvalue by \(\lambda_1 (\Omega)>0\) and by \(B^3\) the Euclidean unit ball in \(\mathbb{R}^3\), the author obtains the Faber-Krahn-type estimate NEWLINE\[NEWLINE\lambda_1 (\Omega) |\Omega|^{2/3} \geq \left( 1 - \left( \frac{2|\Omega|}{9\pi}\right)^{1/3}\right)^2 \lambda_1 (B^3) |B^3|^{2/3},NEWLINE\]NEWLINE provided \(|\Omega| \leq 4\pi/81\). Applying this to the nodal domains of a fixed eigenfunction on \(\mathbb{T}^3\) and combining this with an estimate obtained for the eigenvalue counting function of the Laplacian on \(\mathbb{T}^3\) as usual, the problem is reduced to a finite number of candidate indices \(k\), which may be treated by hand.