Rayleigh’s Principle by Equivalent Problems

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

$$ J\left[ u \right] = \iint {\left[ {p\nabla u \cdot \nabla u + \left( {q - \lambda r} \right){u^2}} \right]} $$

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.