Could someone please tell me the solution to this problem. I have $$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$$ on the square domain $-L<x<L, -L<y<L$ with homogeneous Dirichlet boundary conditions: $$u(L,y)=u(-L,y)=u(x,L)=u(x,-L)=0$$ and further condition $u(0,0)=1$.
I have tried separation of variables but quickly realised that it would only give me the trivial $u=0$.
This is not for some kind of homework, I am solving the heat equation numerically and would just like to know what the analytical steady state solution is so I may compare. Thanks in advance!
There's only one solution to the problem $$\begin{cases} \Delta u =0 & \Omega, \\ u=0 & \partial \Omega \end{cases} $$ when $\Omega$ is a domain with Lipschitz continuous boundary, such as the square. This solution is the identically vanishing one, as you can see in various ways; one of them is observing that the energy integral $$\int_{\Omega} \lvert \nabla u\rvert^2\, dx$$ must vanish.