I'm studying by myself PDE by the Evans' book and I'm trying understand the Second Existence Theorem for weak solutions of elliptic equations and I'm trying understand the steps $1$ and $3$ of the proof (this topic has an image with the theorem and the proof), but there are two points that I didn't understand:
1) Why $(13)$ is true? I think it is used the Lax-Milgram Theorem, but I don't sure because $u$ would be in $U$ and not in $H^1_0(U)$, right? Because $g \in L^2(U)$ and $u$ must be an element of the domain of $g$ by the Lax-Milgram Theorem.
2) How obtain $|| Kg ||_{H^1_0(U)} \leq C ||g||_{L^2(U)}$?
Thanks in advance!
You are correct that you can verify this via Lax-Milgram lemma. I think you have confused $g$ and the linear functional $H^1_0(U)\to \mathbb{R}$ given by $v\mapsto (g,v)$. The domain of that operator is $H^1_0(U)$, not $U$. $U$ is the domain of the functions $u,v,g,f$.
It tells you to refer back to a previous section, which I believe introduces the trace operator, which relates a function to its boundary values and establishes the bound you mention