The main results of this paper is two-fold. Firstly, let \(\mathcal {A}\subset \mathbb {R}^2\) be a planar circular annulus and denote by \(\mathcal {N}_{N}(\mathcal {A},\lambda )\) the counting function of Neumann eigenvalues of Laplacian in \(\mathcal {A}\) . Pólya’s conjecture states that \(\mathcal {N}_{N}(\mathcal {A},\lambda )\ge (4\pi )^{-1}|\mathcal {A}|\lambda ^2\) and Kröger then proved that \(\mathcal {N}_{N}(\mathcal {A},\lambda )\ge (8\pi )^{-1}|\mathcal {A}|\lambda ^2\) . In this paper, we prove that there exists a better estimate for \(\mathcal {N}_{N}(\mathcal {A},\lambda )\) than Kröger’s estimate. Also, we establish better estimates for Neumann eigenvalues of Laplacian in planar annular sectors. Secondly, we study the following boundary mean zero Laplacian eigenvalue problem with constant Neumann boundary data on unit disk \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta w=\kappa w \quad & \hbox {in}\ \mathcal {D}, \\ \frac{\partial w}{\partial \vec {n}}=-\frac{\kappa }{P(\mathcal {D})}\int _{\mathcal {D}}w\,dx \quad & \hbox {on}\ \partial \mathcal {D}, \\ \int _{\partial \mathcal {D}}w\,d\sigma =0, \ \ & \, \end{array}\right. } \end{aligned}\) where \(P(\mathcal {D})\) denotes the perimeter of \(\mathcal {D}\) . By establishing a conclusion similar as the famous Bourget’s hypothesis, we find all the eigenvalues about the above problem. Besides, we investigate the asymptotic distribution of this new type eigenvalue on disk. We prove that the corresponding Pólya’s conjecture for the counting function \(\mathcal {N}_{CN}(\mathcal {D},\lambda )\) fails for \(\lambda \in (0,\infty )\) and then find the precise range on parameter \(\lambda \) that makes \(\mathcal {N}_{CN}(\mathcal {D},\lambda )\ge \frac{1}{4}\lambda ^2\) , i.e., the corresponding Pólya’s conjecture, valid. Also, we establish some asymptotic estimate for this new type of eigenvalues.
