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On exact Pleijel’s constant for some domains. (English) Zbl 1401.35214
A classical theorem of Courant states that any eigenfunction \(\varphi_n\) associated with the \(n\)th eigenvalue \(\lambda_n\) of the Dirichlet Laplacian on a bounded domain \(\Omega \subset \mathbb{R}^N\), \(N\geq 2\), has at most \(n\) nodal domains (that is, connected components of the set \(\{x \in \Omega: \varphi_n (x) \neq 0 \}\)).
Denote by \(\mu(\varphi_n)\) the number of nodal domains of \(\varphi_n\). Then by another well-known theorem, this time due to Å. Pleijel [Commun. Pure Appl. Math. 9, 543–550 (1956; Zbl 0070.32604)], for any bounded domain \(\Omega\) and any choice of \(\varphi_n\), we have \[ Pl(\Omega):=\limsup_{n\to\infty} \frac{\mu(\varphi_n)}{n} < 1. \] This is in marked contrast to the one-dimensional case, where \(\mu(\varphi_n)=n\) for all \(n\geq 1\), by a classical theorem from Sturm-Liouville theory. Very little is known about \(Pl(\Omega)\) for \(\Omega\) in two or more dimensions; even the maximal possible value of \(Pl(\Omega)\), i.e., \(\sup_{\Omega} Pl(\Omega)\), is unknown, and the best known upper bounds are not optimal (see Remark 2.2 of I. Polterovich [Proc. Am. Math. Soc. 137, No. 3, 1021–1024 (2009; Zbl 1162.35005)] or S. Steinerberger [Ann. Henri Poincaré 15, No. 12, 2299–2319 (2014; Zbl 1319.35132)].
In the current paper, the author computes \(Pl(\Omega)\) explicitly for a number of domains; previously, \(Pl(\Omega)\) seems only to have been known for certain rectangles (see, e.g., Proposition 5.1 of B. Helffer and T. Hoffmann-Ostenhof [Contemp. Math. 640, 39–57 (2015; Zbl 1346.35132)]). Here, most notably, the value on the unit disk \(B\) in \(\mathbb{R}^2\) is computed via a fine analysis of Bessel functions and their zeros: \[ Pl(B) = 8\sup_{x>0} \left\{ x(\cos(\theta(x))^2\right\} \simeq 0.46, \] where \(\theta = \theta(x)\) is the unique solution of the transcendental equation \[ \tan \theta - \theta = \pi x \] in the interval \(\theta \in (0,\frac{\pi}{2})\). Other domains covered include circular sectors (under a simplicity assumption on the eigenvalues known to hold in many concrete cases), certain planar annuli, and \(N\)-dimensional hyperrectangles under an incommensurability assumption on the side lengths; the latter generalizes the known two-dimensional results.

MSC:
35P05 General topics in linear spectral theory for PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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