# Homework Help: Solutions to Laplace's equation in Sobolev spave (existence of)

1. Mar 15, 2013

### TaPaKaH

1. The problem statement, all variables and given/known data
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 $u\in W_0^{1,2}(U).$

2. Relevant 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.