Solutions to Laplace's equation in Sobolev spave (existence of)

[TaPaKaH]

Homework Statement

Let $U\subset\mathbb{R}^m$ be a bounded set with smooth boundary $\partial U$.
Consider a boundary value problem $$-\bigtriangleup u=f,\quad u|_{\partial U}=0.$$with $f\in L^2(U)$.
Use the Riesz representation Theorem that the problem has a weak solution $u\in W_0^{1,2}(U).$

Homework Equations

$u\in W_0^{1,2}(U)$ is a weak solution if $\int_U\sum_{k=1}^m u_{x_k}v_{x_k}=\int_U fv$.
It is not allowed to use Lax-Milgram Theorem, but it is hinted that Poincare inequality might be useful.

From what I can see, the exercise will involve various norm estimates, but what I can't see is how these estimates can yield the existence.

 

[lippyka]

The Attempt at a SolutionI don't know how to solve this problem. I understand that the Riesz Representation Theorem states that if there exists a function $v\in W_0^{1,2}(U)$ such that $\int_U fv=\int_U\sum_{k=1}^m u_{x_k}v_{x_k}$ for all $v\in W_0^{1,2}(U)$, then we have a weak solution.But how can we prove the existence of such a function?