Existence of solution of weak solution of Dirichlet Problem

Question

I'm reading "Topics in Functional Analysis and Applications" by S.Kesavan and I have a doubt in existence of weak solution of Dirichlet problem.

There theorem is:

where 3.2.3 is

However, in the statement of Lax-Milgram, for $f \in V$, there exists a $u \in V$. So according to this, for $f \in H^1$, there exists a $u \in H^1$. This doesn't prove what we require.

How does the conclusion follow?

Answer 1

The missing link is the fact that for $f\in L^2(\Omega)$ fixed, the right hand side of (3.2.3) is a bounded linear functional on $H_0^1(\Omega)$. So, by Riesz representation there exists one $\hat f\in H_0^1(\Omega)$ for which

$$ \int_\Omega fv =(\hat f,v)_{H_0^1}. $$

Since the inequalities verified in the text for $a(u,v)$ hold true in $H_0^1(\Omega)$, then you can apply Lax-Milgram in that space and obtain the required result.

Of course, I agree with you in that the proof as shown above seems only to address the case in which $f\in H^1(\Omega)$.