Max Tegmark's library: hadamard

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Harold Shapiro & Max Tegmark


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 wrong! 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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