Help understanding the proof of (sequential weak) coercivity of the Dirichlet integral $I(u)=\int_{\Omega}|\nabla u(x)|^2 dx$

Question

Let $\Omega \subset \mathbb R^n$, open and bounded with Lipschitz boundary

Consider the Dirichlet integral:

$$I(u)=\int_{\Omega}|\nabla u(x)|^2 dx\tag {5.3}$$

and the class of functions

$\Phi= \{ u \in W^{1,p}(\Omega) : u-g \in W^{1,p}_0(\Omega) \}$ with $g \in W^{1,p}(\Omega)$ given.

When proving the existence of a unique minimizer $u $ in that class of functions that satisfies the weak form of the euler Lagrange equations, the first step is to prove that I is sequentially weakly coercive. I am having trouble understanding why the proof of that part in my notes is correct.

You can see below the whole proof .

I have a couple of definitions within a corollary that is used in the proof:

So I think the proof of coercivity should start by considering the assumptions of the definition: Let $\alpha \in \mathbb R$ be ARBITRARY and also consider an ARBITRARY sequence $(u_k)$ sucht that $I(u)\le \alpha$. I have to prove that this arbitary sequence has a weaklyconvergent subsequence in $W^{1,2}(\Omega)$

1. I don't see where they are considering these assumptions in their proof and besides they got a weakly convergent subsequence just for a PARTICULAR sequence (a minimizing sequence), because they used that $I(u_k)$ is bounded to imply that the sequence $u_k$ is bounded. Certainly not all sequences in $W^{1,2}\Omega$ have to be minimizing sequences! Therefore it doesn't look like they are following the definition. Can you help me clarify this?

2. Shouldn't the proof of coercivity end once I found the weakly convergent subsequence? Why do they end it only after using theorem 3.33 (see highlighted sentence)to prove that $u-g \in W^{1,2}\Omega$, and that is that $u \in \Phi$? This doen't seem to have anything to do with the proof of coercivity.

---

**Whole proof **

Quoted theorems:

Answer 1

1. There seems to be a subtle shortcut the author used here to avoid defining too many sequences and repeating arguments.

The main proof here is the proof of the Theorem 5.5, so they're defining a minimizing sequence $(u_k)$ first. Within that proof, they're also proving that $I:\Phi\mapsto\mathbb{R}$ is weakly coercive, and using the same sequence to do so.

However, notice that they don't use the fact that $(u_k)$ is a minimizing sequence in that "subproof". They only use that fact to show that $I(u_k)$ is bounded and therefore satisfies the hypotheses needed to prove weak coercivity, but the minimizing aspect doesn't influence the subproof itself. There's also no assumptions on what that upper bound is, so the "for any $\alpha$" part is still valid.

I can empathize with the frustration of proofs that may be a bit lax with their arguments lol, especially as an introduction to the topic. A clearer way to do it may have been to prove the weak coercivity of $I:\Phi\mapsto\mathbb{R}$ in a separate lemma, but I get that there's also a tricky sweet spot between brevity and clarity.

2. Since the aim here is to show the existence of a minimizer $u$ belonging to $\Phi$, they're specifically trying the show the weak coercivity of $I$ over $\Phi$, not just $W^{1,2}(\Omega)$ in general.