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Problem: Let $ \lambda\in\mathbb{R}, u $ a smooth function, not identically zero, defined on a neighborhood of the unit disc satisfying $ \Delta u+\lambda u = 0 $ in the interior of the unit disc and $u=0$ on the boundary. Show $\lambda > 0$.

I already know how to handle these types of problems by multiplying the PDE by an appropriate function (u here) and integrating using Green's theorem. A detailed outline to essentially the same problem (by user Michael Chen) can be found under my favorites.

My question is whether you can solve this using the Fourier transform (FT) over the unit disc? I ask because in one dimension, the Fourier transform immediately gives the result, but I'm not too familiar with using the FT in R^n. Thanks.

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What do you have in mind when you say in 1D you immediately get the result from FT? On a disk you can do separation of variables and get a series solution in terms of Bessel functions. This will give more precise information on the eigenvalues than just their sign. –  timur Aug 25 '12 at 23:48
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