We find an asymptotic expression for the first eigenvalue of the biharmonic operator on a long thin rectangle. This is done by finding lower and upper bounds which become increasingly accurate with increasing length. The lower bound is found by algebraic manipulation of the operator, and the upper bound is found by minimising the quadratic form for the operator over a test space consisting of separable functions. These bounds can be used to show that the negative part of the groundstate is small. 1.

Abstract. In this paper, we establish sharp inequalities for four kinds of classical eigenvalues on a bounded domain of a Riemannian manifold. We also establish asymptotic formulas for the eigenvalues of the buckling and clamped plate problems. In addition, we give a negative answer to the Payne conjecture for the one-dimensional case. 1.