I want to ask about the following theorem about uniqueness of solution of Dirichlet boundary problem to Laplace equation:
Let $ \varOmega $ be a bounded domain with piecewise wmooth boundary, and let $ f\left(x\right):\overline{\varOmega}\to\mathbb{R} $ and $ \varphi\left(x\right):\partial\varOmega\to\mathbb{R} $ be continuous functions. Then, there exists at most one solution $ u\left(x\right)\in C^{2}\left(\overline{\varOmega},\mathbb{R}\right) $ to the Dirichlet boundary problem for the laplace equation $$ \begin{cases} \varDelta u\left(x\right)=f\left(x\right) & x\in\varOmega\\ u\left(x\right)=\varphi\left(x\right) & \forall\partial\varOmega \end{cases} $$
That is the theorem as it stated in my lecture notes (also there is a proof, which I totally understand). But there is something I dont get.
Consider the Laplace equation where $ \varOmega = (0,L)\subset \mathbb{R} $ where $L>0 $, and $ f(x)= -\lambda\cdot u(x) $ where $\lambda >0 $.
Then $ \partial\varOmega=\left\{ 0,L\right\} $, and assume $ u\left(0\right)=0,\thinspace\thinspace u\left(L\right)=0 $. (That is, $\varphi = 0 $).
So the problem is reduced to ordinary differential equation,and we know that $ \left\{ \sin\left(\frac{\pi n}{L}x\right)\right\} _{n=1}^{\infty} $ are all eigenfunctions of the problem, and specifically, each eigenfunction solves the problem. So how does that fit with the uniquness of solution to the Dirichlet problem?
Suppose you consider the problem with $f(x)=\sin \pi x/L$. Then its unique solution with $\phi=0$ is $u(x)=-L^2f(x)/\pi^2$. There is no contradiction here, because neither $u=0$ or other (multiples of) eigenfunctions are solutions with the same $f$ on the right hand side.