On the eigenvalues and eigenfunctions of elastic plates

✍ Scribed by Åke Pleijel

• Publisher: John Wiley and Sons
• Year: 1950
• Tongue: English
• Weight: 406 KB
• Volume: 3
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/cpa.3160030102

✦ Synopsis

This paper contains some investigations concerning the asymptotic behavior of eigenvalues and eigenfunctions of elastic plates undertaken at the Institute for Advanced Study a t Princeton. The theory is based on Carleman's ingenious iden i~ his papers [l], (21. The necessary estimations of the Green's functions am found by variational methods. For the theory of elastic plates we refer ta Friedricks [4]. We consider the bilinear form where A = a2/dx2 + d2/dy2, u,, = a2U/dx2; u,, = d2u/dx dy and so on; p is a constant, 0 5 p < 1. The integral is extended over a finite open region V of the euclidean xy-plane where x and y are Cartesian coordinates. This region V represents the elastic plate. On account of the inequality for p the quadratic form J ( u , u) 2 0. Let S be the boundary of V and let T be the (finite) set of corner-points of S. lye denote by 7~ = (n, , n,) the outer normal and by s the arc-length of the boundary, oriented counter-clockwise. By formal integrations by parts we find the formula J(u, v) = /" UAAV dx dyu I&) ds + 1 &/dn I,(v) ds s where + The meaning of I a t a point s is -' + I constant K -~ Zi2.