Lower-bound estimates for eigenvalue of the Laplace operator on surfaces of revolution. (Q1432834)

For a surface of revolution \(S\) which is represented as :\(\{ (x, f(x) \cos \theta, f(x) \sin \theta): x \in (\alpha, \beta), 0 \leq \theta \leq 2 \pi \}\) with \(f(x)>0\), \(f \in C^2 (\alpha, \beta)\), the Laplace eigenvalue problem on \(S\) with mixed or Neumann boundary conditions can be separated. Consider the surface area as a new variable, that is, let \(y(x) = 2 \pi \int_\alpha^x w(t) f(t) \,dt\) with \(w(t) = \sqrt{1 + [f'(t)]^2}\), one of the reduced ODE's which has \(x\) as variable can be transformed to a self-adjoint Sturm-Liouville eigenvalue problem with corresponding boundary conditions. Using the variational characterization of the eigenvalues for the Sturm-Liouville problem and two inequalities from \textit{B. A. Troesch} [Arch. Ration. Mech. Anal. 24, 128--140 (1967; Zbl 0152.04802)], under some restriction of \(f\), the author accordingly, for the Laplacian on \(S\), gets respectively a lower bound (which is optimal) for the first eigenvalue with the mixed boundary condition, and a lower bound for the first positive eigenvalue which has odd multiplicity with the Neumann boundary condition. The area \(A\) of \(S\) is involved in both estimates. The restriction of \(f\) is due to that of Troesch's inequalities.