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Where can I find some information about: find the eigen values and eigen vectors $(\lambda,u)$ of the Sturm-Liouville problem

-$div(\rho^{\alpha+1}\nabla u)=\lambda\rho^\alpha u$

where $\alpha>-1$ and $\rho(x,y)=1-x^2-y^2$?

I'm working on $D=\{(x,y)\in\mathbb{R}^2:\;x^2+y^2\leq1\}$.

Thanks!

hardmath
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yemino
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  • Perhaps you mean that you are working on the unit disk in $\mathbb{R}^2$? Also, what is $\rho$? The radius? – hardmath Jun 11 '14 at 23:33
  • thanks @hardmath , I added the missing information. – yemino Jun 12 '14 at 00:27
  • The unit disk is the finite region bounded by the unit circle, so I feel sure that you meant to pose this PDE on that region (rather than on its boundary, where $\rho = 0$ according to your formula). – hardmath Jun 12 '14 at 00:31
  • (edited) do you know where can I find information about this kind of problems? – yemino Jun 12 '14 at 00:45
  • I see it is a self-adjoint type of PDE, when converted to divergence form. Are you more interested in the specific spectrum of this operator, or in some references about similar PDE operators? – hardmath Jun 12 '14 at 01:18
  • To begin, I am interested in finding the values ​​and eigenvectors. I suppose it is necessary to transform the pde to polar form. – yemino Jun 12 '14 at 01:27

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