Abstract
Rayleigh’s Principle provides an inequality which gives upper bounds to an eigenvalue of a differential equation by evaluating a ratio of quadratic functionals with any function from a prescribed class. It also shows that the value of the functional evaluated with the eigenfunction is exactly the eigenvalue. This paper shows how minimizing the functional
by a modification of Carath€odory’s equivalent-problems method yields Rayleigh’s Principle for the partial-differential-equation (PDE) eigenvalue problem ▽ · (p▽u) – (q – λr)u = 0 on D, u = 0 on ∂D. The approach leads to a Hamilton-Jacobi equation for a vector variable, and seeking special forms of solution to this leads to a scalar PDE \( \nabla \cdot \sigma + \frac{1}{p}\sigma \cdot \sigma = q - \lambda r \) for the vector variable σ(x,y). This PDE is an obvious generalization of a Riccati ordinary differential equation, and it can be linearized by the transformation \( \sigma = p\frac{{\nabla w}}{w} \) . The resulting linear PDE is the original eigenvalue PDE, which has a nonvanishing solution by hypothesis. This guarantees the existence of a “nice” equivalent problem which immediately yields Rayleigh’s Principle, both the upper-bound and equality parts. The proof given here is for the two-dimensional case, but the specialization to the one-dimensional case and the generalization to the more-than-two-dimensional cases are immediate.
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References
C. Carathéodory, Variationsrechnung und partielle Differential-gleichungen erster Ordnung, Teubner, Leipzig, 1935 (English translation, Holden-Day, San Francisco, 1966 ).
H. Rund, The Hamilton-Jacobi theory in the calculus of variations, Van Nostrand, London, 1966 (reprinted with corrections by Krieger, Huntington, N.Y., 1973 ).
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D. R. Snow, Transversality and natural boundary conditions by equivalent problems in calculus of variations, in Calculus of variations and control theory, edited by David L. Russell, Academic Press, New York, 1976, 391–404.
D. R. Snow, Using equivalent problems to solve Bolza’s problem of the calculus of variations, Annual Meeting of the American Mathematical Society, San Antonio, Texas, 22–25 January,1976; abstract in Notices of the American Mathematical Society, 23 (1976), A-169.
D. R. Snow, A new proof for Rayleigh’s principle for eigenvalue approximations, Annual Meeting of the American Mathematical Society, Washington, D. C., 21–26 Januaxy,1975; abstract in Notices of the American Mathematical Society, 22 (1975), A-198.
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Snow, D.R. (1978). Rayleigh’s Principle by Equivalent Problems. In: Beckenbach, E.F. (eds) General Inequalities 1 / Allgemeine Ungleichungen 1. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 41. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5563-1_23
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DOI: https://doi.org/10.1007/978-3-0348-5563-1_23
Publisher Name: Birkhäuser, Basel
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