Vibrational modes of circular free plates under tension

References

• Rayleigh , JWS . 1945 . The Theory of Sound , New York : Dover Pub . (Re-publication of the 1894/96 edition)
   Google Scholar
• Faber , G . 1923 . Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisfömige den tiefsten Grundton gibt . Sitzungberichte der mathematisch-physikalischen Klass der Bayerischen Akademie der Wissenschaften zu München Jahrgang , : 169 – 172 .
   Google Scholar
• Krahn , E . 1925 . Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises . Math. Ann. , 94 : 97 – 100 .
   Google Scholar
• Krahn , E . 1926 . Über Minimaleigenschaften der Kugel in drei und mehr Dimension . Acta Comm. Univ. Tartu , A9 : 1 – 44 . [English translation: Minimal properties of the sphere in three and more dimensions, Edgar Krahn 1984–1961: A Centenary Volume, Ü. Lumiste and J. Peetre, editors, IOS Press, Amsterdam, 1994, Chapter 11, 139–174]
   Google Scholar
• Szegő , G . 1950 . On membranes and plates . Proc. Nat. Acad. Sci. , 36 : 210 – 216 .
   PubMed Web of Science ®Google Scholar
• Szegő , G . 1958 . Note to my article ‘On membranes and plates’ . Proc. Nat. Acad. Sci. (USA) , 44 : 314 – 316 .
   PubMed Web of Science ®Google Scholar
• Weinberger , HF . 1956 . An isoperimetric inequality for the N-dimensional free membrane problem . J. Rational Mech. Anal. , 5 : 633 – 636 .
   Web of Science ®Google Scholar
• Talenti , G . 1981 . On the first eigenvalue of the clamped plate . Ann. Mat. Pura Appl. (Ser. 4) , 129 : 265 – 280 .
   Web of Science ®Google Scholar
• N.S. Nadirashvili. New isoperimetric inequalities in mathematical physics. in Partial differential equations of elliptic type (Cortona, 1992), Symposia Mathematica XXXV, Cambridge University Press, Cambridge, 1994, pp. 197–203.
   Google Scholar
• Nadirashvili , NS . 1995 . Rayleigh's conjecture on the principal frequency of the clamped plate . Arch. Rational Mech. Anal. , 129 : 1 – 10 .
   Web of Science ®Google Scholar
• Ashbaugh , MS and Benguria , R . 1995 . On Rayleigh's conjecture for the clamped plate and its generalization to three dimensions . Duke Math. J. , 78 : 1 – 17 .
   Web of Science ®Google Scholar
• Ashbaugh , MS and Laugesen , RS . 1996 . Fundamental tones and buckling loads of clamped plates . Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) , 23 : 383 – 402 .
   Google Scholar
• M.S. Ashbaugh, R. Benguria, and R.S. Laugesen. Inequalities for the first eigenvalues of the clamped plate and buckling problems. General inequalities 7 (1995), pp. 95–110.
   Google Scholar
• Leissa , AW . 1969 . Vibration of Plates NASA SP-160, Office of Technology Utilization , Washington, DC : NASA, US Government Printing Office .
   Google Scholar
• Nazarov , SA and Sweers , G . 2007 . A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners . J. Differential Equations , 233 : 151 – 180 .
   Web of Science ®Google Scholar
• L.M. Chasman. An isoperimetric inequality for fundamental tones of free plates. Preprint, arXiv:1004.3318 [math.SP].
   Google Scholar
• Verchota , GC . 2005 . The biharmonic Neumann problem in Lipschitz domains . Acta Math. , 194 : 217 – 279 .
   Web of Science ®Google Scholar
• Kawohl , B , Levine , HA and Velte , W . 1993 . Buckling eigenvalues for a clamped plate embedded in an elastic medium and related questions . SIAM J. Math. Anal. , 24 : 327 – 340 .
   Web of Science ®Google Scholar
• Payne , LE . 1958 . Inequalities for eigenvalues of supported and free plates . Quart. Appl. Math. , 16 : 111 – 120 .
   Google Scholar
• L.M. Chasman. Isoperimetric problem for eigenvalues of free plates, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2009.
   Google Scholar
• Weinstock , R . 1952 . Calculus of Variations , New York : McGraw-Hill Book Company Inc. . (Reprint of 1952 edition.)
   Google Scholar
• M. Abramowitz and I.A. Stegun (eds.), Handbook of Mathematical Functions, Applied Mathematics Series, Vol. 55, National Bureau of Standards, Washington, DC, 1964 (Reprinted by Dover, New York, 1965).
   Google Scholar
• Lorch , L and Szego , P . 1994 . Bounds and monotonicities for the zeros of derivatives of ultraspherical Bessel functions . SIAM J. Math. Anal. , 25 : 549 – 554 .
   Web of Science ®Google Scholar