Nodal domains in the square -- the Neumann case (Q2801888)

Let us furnish some basic definitions and properties, then we state the goal of the article. Let \(u\) be an eigenfunction of \(-\Delta_\Omega\), the positive Laplacian operator on \(\Omega\), a regular bounded domain. Nodal domains of \(u\) correspond of the connected components in the complement of \(u^{-1}(\{0\})\) in \(\Omega\), and their number is denoted by \(\mu(u)\). Courant's theorem states that \(\psi_k\), an eigenfunction associated to the \(k\text{th}\) eigenvalue of \(-\Delta_\Omega\), has at most \(k\) nodal domains, and Courant-sharp theorem states that \(\mu(\psi_k)=k\). The purpose of the article is to consider \(\Omega\) a square with Neumann boundary conditions, thus in this situation the corresponding eigenvalues are given explicitly, i.e., \(n^2+m^2\) such that \((n,m)\) are integers numbers, the authors state that \(k\in \{1,2,4,5,9\}\) (Theorem 1.1). The cases \(\mu(\psi_1),\mu(\psi_2),\) and \(\mu(\psi_5)\) are treated in Lemma 4.2, and for the cases \(\mu(\psi_4),\mu(\psi_9)\), see Lemma 4.4. The proofs are endowed with pleasant plots. The cases of non Courant sharpness are also listed. The authors recall that a square with Dirichlet boundary conditions has already been conducted.