### Max Tegmark's library: hadamard

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described below.
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##
AN ELEMENTARY PROOF THAT THE BIHARMONIC
GREEN FUNCTION OF AN ECCENTRIC ELLIPSE CHANGES SIGN

### Authors:

Harold Shapiro & Max Tegmark
### Abstract:

P. R. Garabedian showed in 1951 that the Green
function for the
biharmonic boundary value problem with vanishing Dirichlet data
changes sign in case the domain is a sufficiently eccentric
ellipse. This refuted a conjecture made by
J. Hadamard
in 1908. The proof of Garabedian was based on kernel functions;
the present note gives an elementary proof.
### Reference info:

Published in Soc. Ind. App. Math. Rev., **36**, 99-101 (1994)
### Comment for physicists:

If a horizontal rubber sheet is strapped into a
rigid frame of some arbitrary shape
(for instance, a circle would make thie problem
correspond to that of a drum skin on a drum), then when all oscillations
have died down, its *z*-coordinate
*z(x,y)* in response to some weak applied force field
*f(x,y)* is of course
given by the Poisson equation *Dz=f*,
where the differential operator *D* is the Laplacian.
It is intuitively obvious that if we press up on any one point on the
rubber sheet, then no part of the sheet will move down,
i.e., *z(x,y)*>=0.
This physical property corresponds to a well-known
mathematical property of the Laplacian: its
Green
function
(operator inverse) is positive definite.
In the above-mentioned paper, we treat the corresponding
problem where *D* is the **square**
of the Laplacian, the so-called bi-Laplacian.
Physically, this corresponds to replacing the rubber sheet with a
rigid metal plate.
The famous mathematician
Hadamard
once conjectured that also
this Green function was positive definite, i.e., that
if you press upwards somewhere on the plate, this could never cause a
downward deflection anywhere. But he was
!
His conjecture was refuted long ago, but using quite complicated
methods. Here we refute the conjecture by showing that there is
quite a trivial
counterexample, where the metal sheet has the shape of a rather
oblong ellipse.

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This page was last modified July 1, 1998.
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