Proof of the Payne-Pólya-Weinberger conjecture

In 1955 and 1956 Payne, Pólya, and Weinberger considered the problem of bounding ratios of eigenvalues for homogeneous membranes of arbitrary shape [PPW1, PPW2]. Among other things, they showed that the ratio k2IK °f * ^ r s t t w o eigenvalues was less than or equal to 3 and went on to conjecture that the optimal upper bound for A2/Aj was its value for the disk, approximately 2.539. It is this conjecture which we establish below. Since 1956 various authors have attempted to prove the conjecture of Payne, Pólya, and Weinberger and some have been able to improve upon the constant 3. Specifically, Brands [Br] in 1964 obtained the value 2.686, de Vries [dV] in 1967 obtained 2.658, and Chiti [Ch2] in 1983 obtained 2.586. In addition, Thompson [Th] gave the natural extension of the PPW argument to dimension n, obtaining

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