# Theorems and Lemma

 The following theorems and lemma are taken from Theory and Practice of Finite Elements by A. Ern and J.L. Guermond.

$\Omega$ is a (measurable) open set of $\mathbb{R}^d$ with boundary $\partial \Omega$. Whenever it is well-defined, its outward normal is denoted by $n$.

## 1. Embedding and Compacity

Proposition B.39

Let $\Omega$ be an open bounded set. Then, for $1 \leq p<q \leq +\infty$, the embedding $L^q(\Omega) \subset L^p(\Omega)$ is continuous.

Proof. This is a consequence of Hölder’s inequality.

One of the key arguments in the embedding theory is the following:

Theorem B.40 (Sobolev).

Let $1 \leq p<d$ and denote by $p^*$ the number such that $\frac{1}{p^*}=\frac{1}{p}-\frac{1}{d}$. Then,

$\begin{equation*} \exists c=\frac{p^*}{1^*}, \forall u \in W^{1, p}\left(\mathbb{R}^d\right), \quad\|u\|_{L^{p^*}\left(\mathbb{R}^d\right)} \leq c\|\nabla u\|_{L^p\left(\mathbb{R}^d\right)} . \end{equation*}$

Proof. See [MaZ97, p. 32], [Sob63, §I.7.4], or [Bre91, p. 162].

Corollary B.41.

Let $1 \leq p, q \leq+\infty$. The following embeddings are continuous:

$\begin{equation*} W^{1, p}\left(\mathbb{R}^d\right) \subset L^q\left(\mathbb{R}^d\right) \text { if }\left\{\begin{array}{l} \text { either } 1 \leq p<d \text { and } p \leq q \leq p^*, \\ \text { or } p=d \text { and } p \leq q<+\infty . \end{array}\right. \end{equation*}$

Proof. See [MaZ97, p. 34], [Sob63, §I.8.2], or [Bre91, p. 165].

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.

## 2. 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) $\square$