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In the Dirichlet problem if nodal lines do not touch $\partial\Omega$ (unit disk), what happens to the eigenvalues?

Thanks for help.

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Why do you need to know? Please have a read of – David Roberts May 2 '13 at 1:13
If you are assuming $\Omega$ is a disk then the eigenvalues and eigenmodes can be computed explicitly. If I remember correctly, if the nodal lines do not touch the boundary of $\Omega$ then the eigenvalues are simple. (this is just for the disk) – Beni Bogosel May 25 '14 at 22:50

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