# Theorems and Lemma

 The following theorems and lemma are taken from Theory and Practice of Finite Elements by A. Ern and J.L. Guermond.
Theorem B.46 (Rellich-Kondrachov)

Let $1 \leq p \leq+\infty$ and let $s \geq 0$. Let $\Omega$ be a bounded open set having the $(s, p)$ - extension property. The following injections are compact:

(i) If $p \leq d, W^{s, p}(\Omega) \subset L^{q}(\Omega)$ for all $1 \leq q<p^{*}$ where $\frac{1}{p^{*}}=\frac{1}{p}-\frac{s}{d}$

(ii) If $p>d, W^{s, p}(\Omega) \subset \mathcal{C}^{0}(\bar{\Omega})$

Proof. See [MaZ97], [BrS94, Chap. 1], or [Bre91, Chap. 8]. $\square$

 Theorem B.46 (Rellich-Kondrachov) states a very useful compacity result.

## 1. Poincaré-like inequalities

Lemma B.61 (Poincaré).

Let $1 \leq p<+\infty$ and let $\Omega$ be a bounded open set. Then, there exists $c_{p, \Omega}>0$ such that

$\forall v \in W_{0}^{1, p}(\Omega), \quad c_{p, \Omega}\|v\|_{L^{p}(\Omega)} \leq\|\nabla v\|_{L^{p}(\Omega)} ]$

For $p=2,$ we denote $c_{\Omega}=c_{2, \Omega}$

Proof. We only give the proof for $p<d$. Let $\tilde{v} \in W^{1, p}\left(\mathbb{R}^{d}\right)$ be the zeroextension of $v ;$ see Proposition B.48. Theorem B.40 implies $\|\tilde{v}\|_{L^{p^{*}}\left(\mathbb{R}^{d}\right)} \leq$ $c\|\nabla \tilde{v}\|_{L^{p}\left(\mathbb{R}^{d}\right)} .$ since $\Omega$ is bounded and $p^{*} \geq p,$ we infer $\|v\|_{L^{p}(\Omega)}=\|\tilde{v}\|_{L^{p}\left(\mathbb{R}^{d}\right)} \leq$ $c\|\tilde{v}\|_{L^{p *}\left(\mathbb{R}^{d}\right)},$ yielding Lemma B.61 (Poincaré).

Lemma B63

Let $1 \leq p<+\infty$ and $\Omega$ be a bounded connected open set having the $(1, p)$ - extension property. Let $f$ be a linear form on $W^{1, p}(\Omega)$ whose restriction on constant functions is not zero. Then, there is $c_{p, \Omega}>0$ such that

$\forall v \in W^{1, p}(\Omega), \quad c_{p, \Omega}\|v\|_{W^{1, p}(\Omega)} \leq\|\nabla v\|_{L^{p}(\Omega)}+|f(v)|$

Proof. Use the Petree-Tartar Lemma. To this end, set $X=W^{1, p}(\Omega), Y=$ $\left[L^{p}(\Omega)\right]^{d} \times \mathbb{R}, Z=L^{p}(\Omega),$ and $A: X \ni v \mapsto(\nabla v, f(v)) \in Y .$ Owing to Lemma B. 29 and the hypotheses on $f, A$ is continuous and injective. Moreover, the injection $X \subset Z$ is compact owing to Theorem B.46 (Rellich-Kondrachov) $\square$