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.
