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I'm not a mathematician, I may be inaccurate ; sorry for that ! I have two questions.

First, I need some help to clarify the method about solving and homogeneous PDE. I have read somewhere (and my advisor said) that finding the eigenvalues of an differential operator can be helpful to construct the solutions of the homogeneous PDE associated to this operator.

What does that mean ? Is it true for all order of differential operators ? Or only for second order ?

On another hand, I'm working on solving the biharmonic equation $\nabla^4 f = 0$ on some strange subspaces in $\mathbb{R}^2$. My advisor said that on the whole plane, the smaller eigenvalue associated to $\nabla^4$ will be zero. Is it true ? Why ?

Thanks for your answer. I already search references about all of that on Internet but didn't find anything suitable for my problem.

Have a nice day ! :-)

Edit : Solving $\nabla^4 f = 0$ on the plane and on the sphere will have different solutions ? Is it relevant to study the comportment of the solution on the plane to infer something about the solution on the sphere ? And why is my first sentence "Hello everyone" deleted when I post my question ?

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    Okay, thanks for the comments and for the reference ! :-)2012-03-12

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