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I have a question. If we want to solve heat equation on torus or any bounded domain in $\mathbb{R}^n$, we can use the method of separation of variables and the question of getting the eigenvalue and eigenfunctions of the Laplacian arises. So my question is, how we get the eigenvalues and eigenfunctions of Laplacian on the torus(in general, n torus)? And what if the domain is the rectangle? Thanks for any help!

violin
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  • In a rectangle, you separate variables. The strategy is essentially the same on the torus. Use the descrete Fourier transform. – User8128 Mar 19 '16 at 15:32
  • How we define the discrete Fourier transform in a rectangle? – violin Mar 19 '16 at 15:42
  • Expand $f$ in terms of sines and cosines. This is how Fourier series come up. This works because sine and cosine with the correct arguments are eigenfunctions of the Laplacian, which is a self-adjoint operator and the eigenfunctions of a self-adjoint operator form a basis for the solution space. – User8128 Mar 19 '16 at 16:00
  • @User8128 Thanks for explaining the rectangle case. So how we find the eigenfunctions of the n torus? If we expand f in terms of its Fourier series, in what sense does the series converge? And I compute it then I don't how to solve it. – violin Mar 19 '16 at 16:13
  • The torus and rectangle can be identified with each other in a natural way so the strategy is the exact same. Here is a reference: https://www.math.washington.edu/~greenbau/Math_555/Course_Notes/555notes6.ps_pages.pdf – User8128 Mar 19 '16 at 16:27
  • Thanks for the explaining! – violin Mar 19 '16 at 16:35
  • For regions with boundary, there are boundary conditions. For your case you require periodic conditions. Both types of conditions are valid when using separation of variables. – Disintegrating By Parts Mar 22 '16 at 13:22

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