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Math 5440-1
5440-1 home page
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1) Rayleigh-Ritz variational characterization of Laplace eigenvalues. See Friday's notes
dec3.pdf
You can also characterize the nth eigenvalue as the minimum over all collections of n-dimensional subspaces of C1 functions having zero boundary data of the maximum Rayleigh quotient for such subspaces. (mini-max characterization) Notice this implies that larger domains have lower natural frequencies. An interesting question is the extent to which you can "hear the shape of a drum"...
http://en.wikipedia.org/wiki/Hearing_the_shape_of_a_drum
2) The example of a square or rectangle, also on Friday's notes. Notice we can deduce we have a complete eigenbasis because we prove the closure of the span of our collection includes all differentiable functions with zero boundary data...any missing eigenspace would have a unit eigenfunction perpendicular to this closed space, and so wouldn't be approximatable in the L2 norm.
Play with square eigenvalues and eigenfunctions via the wave equation here:
http://www.falstad.com/membrane
http://www.falstad.com/circosc disk domain.
http://www.falstad.com/mathphysics.html more applets
3) Separating variables for heat and wave equations in radial and spherical coordinates, in R2, R3, Rn. This involves the spherical harmonics in the non-radial directions:
http://en.wikipedia.org/wiki/Spherical_harmonics
http://en.wikipedia.org/wiki/Spherical_harmonics#Higher_dimensions
Some of the details are explained further in these notes:
dec6.pdf
And it involves Bessels differential equation in the radial direction:
http://en.wikipedia.org/wiki/Bessel_function