Asymptotic lower bounds for the fundamental frequency of convex membranes

Variational methods of the Rayleigh-Ritz type are frequently used to approximate λ. They always yield upper bounds for λ, and the upper bounds can be made arbitrarily close. Another common practical method of approximating λ is to calculate the least eigenvalue λh of a suitably chosen finite-difference operator Δn over a network with small mesh width h. For one choice of Δh it was shown by Courant, Friedrichs, and Lewy [3, p. 57] without details that λnr+λ as /&->0. For convex regions R of a special polygonal form the author has shown [4] that a special case of (11) below is valid for a common choice of Ah, and hence that λh is asymptotically a lower bound for λ as &-»0. For an unusual finite-difference approximation to problem (1) when R is the union of squares of the network, Polya [12] has found that λ^>λ for all h, and also for the higher eigenvalues. The author knows of no other study of the sign or order of decrease of λ-λh to 0. In the present paper the investigation of [4] is extended to a much wider class of regions: those with piecewise analytic boundary curves and convex corners. The new theorems are stated and proved in §§ 3 and 4. Theorem 2 contains the theorem of [4] as a special case. Lemmas used in the proof of Theorem 1 are given in § 5. Identity (31) of Lemma 7 is interesting in itself.