Vol. 75, No. 1, 1978

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Duffin’s function and Hadamard’s conjecture

Mitsuru Nakai and Leo Sario

Vol. 75 (1978), No. 1, 227–242

The purpose of the present paper is to apply our “beta densities” to Hadamard’s conjecture on the constant sign of the biharmonic Green’s function of a clamped plate. In particular, we will examine in detail Duffin’s function w from our view point of beta densities. We will show that w is a potential of Δ2w 0 with respect to the Green’s kernel of a clamped plate. As a consequence, the Green’s function of the clamped infinite strip is of nonconstant sign along with w. On the other hand, we show using beta densities that the Green’s function of any clamped bounded subregion exhausting the strip tends to that of the clamped strip and, therefore, takes on both positive and negative values. Since the infinite strip can be exhausted by ellipses, we have at once, without carrying out any numerical computations, the Garabedian result: a sufficiently eccentric ellipse is a counterexample to Hadamard’s conjecture. Since the strip can also be exhausted by rectangles, we can add a sufficiently long rectangle to counterexamples to Hadamard’s conjecture. If this may be called a new example, then countless “new” examples can be produced by exhausting the strip by “new” subregions.

Mathematical Subject Classification 2000
Primary: 31A30
Secondary: 35J40
Received: 12 November 1976
Revised: 2 February 1977
Published: 1 March 1978
Mitsuru Nakai
Leo Sario