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I need to calculate the first eigenvalue of Laplacian $-\Delta$ for unit square $(0,1)\times (0,1)$. As a hint I got that consider the function $u(x, y)=\sin(\pi x)\sin(\pi y)$. All tools I have are the eigenvalue equation $-\Delta u=\lambda u$ in $\Omega$, $u=0$ on $\partial\Omega$ and the Rayleigh quotient $$Q(w)=\frac {\int_\Omega |\nabla w|^2}{\int_\Omega w^2},$$ and I know that $\min Q$ is the smallest eigenvalue. I calculated the eigenvalue corresponding to the hint and got $\lambda=2\pi^2$. Somehow I should show that all the other eigenvalues are bounded below by $2\pi^2$. I tried some integration tricks but they were of no use. Hints/help is welcome.

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  • $\begingroup$ This problem is some solving by showing the inverse of $-\Delta$ is a compact operator $\endgroup$ – Guy Fsone Oct 11 '17 at 15:36
  • $\begingroup$ @GuyFsone we haven't used any functional analytic tools in the class so there must be an easier way $\endgroup$ – Infinitebig Oct 11 '17 at 15:50
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you can try to solve to solve this problems by variable separation you can put $u(x,y)=f(x)\times g(y)$ you prjectes this in your equation you find solution with $\lambda$ unkown and you can use the the boundary condition to find the $\lambda$ it's easy juste start

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    $\begingroup$ but that's just one class of solutions to the problem, how do I know that every solution is of the form $u(x, y)=f(x)g(y)$? $\endgroup$ – Infinitebig Oct 11 '17 at 15:59
  • $\begingroup$ the sulotion of this problems is unique so you can write the solution as you wich but your function should verify the equation $-\Delta u=\lamba u$ and the boundary condition $\endgroup$ – Bernstein Oct 11 '17 at 16:10
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    $\begingroup$ And how do you know that there are no other functions satisfying the eigenvalue equation for smaller $\lambda$? $\endgroup$ – amsmath Oct 11 '17 at 16:11
  • $\begingroup$ solution is not necessarily unique since there might be other functions corresponding to the same eigenvalue. the first eigenspace is though 1-dimensional so the functions are equal up to constant. $\endgroup$ – Infinitebig Oct 11 '17 at 16:13
  • $\begingroup$ the solutions of the problemes are $(u,\lambda)\in H^[1}_0(\Omega)\times \mathbb{R}$ $\endgroup$ – Bernstein Oct 11 '17 at 16:16

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