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In the picture below, why is $\lambda(g_{ij})$ the first eigenvalue of $-4\Delta+R$?

The first eigenvalue is strange to me. After reading the wiki about it, I find

$$\lambda_1=\inf\limits_{u\ne 0}\frac{\int_\Omega|\nabla u|^2}{\int_\Omega|u|^2}$$

where $u$ is the solution to $$ \begin{cases} \Delta u + \lambda u = 0 & \text{in $\Omega$} \\ \left. u\right|_{\partial \Omega} = 0 \end{cases} $$

But seemingly, it is not the first eigenvalue in pictures 1,2 and 3. Also, why is the first eigenfunction $u_0>0$ in picture 3 bigger than $0$?

Pictures 1, 2 and 3 are from the 203rd page of this paper.

enter image description here enter image description here enter image description here

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  • $\begingroup$ Do you really need Picture 1? You could write the Dirichlet problem yourself. $\endgroup$ – Silvia Ghinassi Nov 24 '15 at 14:30
  • $\begingroup$ @SilviaGhinassi Sorry, I don't know how to write the big $\{$ $\endgroup$ – lanse7pty Nov 25 '15 at 1:10
  • $\begingroup$ \begin{cases} \Delta u + \lambda u = 0 & \text{in $\Omega$} \\ \left. u\right|_{\partial \Omega} = 0 \end{cases} between double $. $\endgroup$ – Silvia Ghinassi Nov 25 '15 at 1:22
  • $\begingroup$ @SilviaGhinassi Thanks ,I have edit it . $\endgroup$ – lanse7pty Nov 25 '15 at 1:27

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