First eigenvalue of $-\Delta$ in unit square $(0,1)\times (0,1)$

Question

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.

Answer 1

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