Sur la fréquence fondamentale d'une membrane vibrante: évaluations par défaut et principe de maximum

Summary

The starting point of the present paper is a simple method ofPayne-Weinberger to obtain lower bounds for the first eigenvalue of a membrane through comparison with the Rayleigh principle for homogeneous strings. A generalization of this idea to auxiliary non-homogeneous strings leads to a maximum principle (7) (of which an inequality ofBarta-Pólya is a special case) or (7′); in its form (8), it was essentially known (although with some unnecessary restrictive hypotheses) toM. H. Protter in 1958. — This principle is closely related to theThomson principle of boundary value problems; in particular, the allowed discontinuities of the concurrent vector fields (in the formulation (7′)) are the same as inThomson's principle.—Someapplications of the principle to the calculation of rather sharp lower bounds are indicated, as well as an intuitivephysical interpretation (in terms of variation of masses and constraints). It is valid inthree dimensions as well; we indicate its extension toSchrödinger's equation.