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ES 240 Problem Set #8, Problem #20 - Green's function of biharmonic operator is not positive definite

Professor Vlassak mentioned that last year every single person did a finite element project.  He said he wanted to see more theory projects, so I decided to take him up on that.

I was browsing around one day and happened upon an article that explained that while the Green's function of the laplacian was positive definite, the biharmonic operator's Green's function is not.  Physically, this has significance. 

Poisson's equation can be used to model the deformation of a thin elastic membrane due to an applied force.  Intuitively, it makes sense that if the membrane is pushed from one direction, every part of it will deflect in that direction.  That this in fact occurs is a result of the positive definiteness of the Green's function of the laplacian.   At one time it was conjectured that the same should be the case for the biharmonic operator (laplacian of the laplacian).  In fact, this is not the case.

What this means is that there are cases when pushing on a plate will cause some portion of the plate to move back towards you!  Again, this results from the non-positive definiteness of the Green's function of the biharmonic operator.

The "classic" case appears to be an elliptical plate.  If the ellipse is eccentric enough, then this curious behavior can occur.  For my project I am going to learn some things about Green's functions and their properties.  I will also learn how to solve for the deflection of an elliptical plate to illustrate this behavior.

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