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On two conjectures in the fixed membrane eigenvalue problem. (English) Zbl 0272.35058

MSC:
35P99Spectral theory and eigenvalue problems for PD operators
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
References:
[1]R. Courant andD. Hilbert,Methods in Mathematical Physics, vol. 1. Interscience, New York (1953).
[2]E. Hopf.A Remark on Linear Elliptic Differential Equations of Second Order. Proc. Amer. Math. Soc.3, 291–293 (1952). · doi:10.1090/S0002-9939-1952-0050126-X
[3]L. E. Payne,Isoperimetric Inequalities and their Applications, SIAM Review,9, 453–488 (1967). · Zbl 0154.12602 · doi:10.1137/1009070
[4]L. E. Payne andI. Stakgold,On the Mean Value of the Fundamental Mode in the Fixed Membrane Problem, J. Appl. Anal., to appear.
[5]L. E. Payne andH. F. Weinberger,A Faber-Krahn Inequality for Wedge Like Domains, J. Math. Phys.39, 182–188 (1960).
[6]G. Pólya andG. Szegö,Isoperimetric Inequalities in Mathematical Physics. Annals of Math. Studies #27, Princeton Univ. Press, Princeton (1951).