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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 mathoverflow.net/howtoask –  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 at 22:50
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