Propriétés de convexité du type de Weyl pour des problèmes de vibration ou d'équilibre
Abstract
General inequalities of type (5) for eigen values were established as early as 1912 by H. Weyl [11] through consideration of integral equations; he used them for studying the asymptotic behaviour of λ_{ n } when n→∞. These inequalities seem to have been forgotten, as some of them were independently rediscovered by several authors (see [4], p. 314 and [10], pp. 483 484). —Such inequalities are here derived and applied in various ways to special problems (vibrating strings, rods, membranes and plates). Explicit lower bounds for λ_{1} are obtained in a somewhat similar manner in terms of the Green's function. For Schrödinger's equation (15'), the very simple inequalities (17') are found. Analogous inequalities are established for a class of equilibrium problems. Many of the inequalities thus obtained express convexity properties (cf. the Postscriptum above and the paper by Pólya and Schiffer quoted there).
 Publication:

Zeitschrift Angewandte Mathematik und Physik
 Pub Date:
 July 1961
 DOI:
 10.1007/BF01591281
 Bibcode:
 1961ZaMP...12..298H