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?