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