# Theorems and Lemma

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

From now on, $\Omega$ 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 of Proposition B.39

This is a result known as the embedding of Lebesgue spaces. It states that for an open bounded set $\Omega$ and $1 \leq p < q \leq +\infty$, the space $L^q(\Omega)$ is continuously embedded in $L^p(\Omega)$. This is a consequence of Holder’s inequality.

To provide some context, $L^p$ spaces are function spaces defined using a natural generalization of the $p$-norm for finite-dimensional vector spaces. They are important in the theory of partial differential equations, Fourier analysis, and many other areas of mathematics.

The result essentially says that if a function is in $L^q(\Omega)$ (i.e., it is 'q-integrable'), then it is also in $L^p(\Omega)$ (i.e., it is 'p-integrable'), and the 'p-integrability' is a stronger condition than 'q-integrability' when $p < q$. The inequality $1 \leq p < q \leq +\infty$ is crucial here because for $p > q$, the embedding $L^p(\Omega) \subset L^q(\Omega)$ is not true in general.

The continuous embedding means that not only every function in $L^q(\Omega)$ is also in $L^p(\Omega)$, but the $L^p$-norm of any such function is bounded by the $L^q$-norm of the function times a constant. This constant does not depend on the function but may depend on the set $\Omega$.

Let’s assume that $f \in L^q(\Omega)$, and let’s show that $f \in L^p(\Omega)$ and that the embedding is continuous. Here $\Omega$ is an open bounded set, and $1 \leq p < q \leq +\infty$.

For any $f \in L^q(\Omega)$, we have by Hölder’s inequality for $r = q/p > 1$ and $r' = r/(r-1)$ (so that $1/r + 1/r' = 1$):

$\|f\|p = \left(\int\Omega |f|^p dx\right)^{1/p} = \left(\int_\Omega |f|^p \cdot 1 dx\right)^{1/p} \leq \left(\int_\Omega |f|^q dx\right)^{1/r} \left(\int_\Omega dx\right)^{1/r'} = \|f\|_q^{p/q} |\Omega|^{1/p},$

where $|\Omega|$ is the measure of the set $\Omega$, i.e., the volume of $\Omega$ in the case where $\Omega$ is a subset of $\mathbb{R}^n$.

So we have shown that $f \in L^p(\Omega)$ and the embedding is continuous because $\|f\|_p \leq C \|f\|_q$ with $C = |\Omega|^{1/p}$.

This completes the proof. The key point here is that Hölder’s inequality gives us a way to compare the $L^p$-norm and the $L^q$-norm of a function. It allows us to say that if a function is 'q-integrable' (i.e., in $L^q(\Omega)$), then it is also 'p-integrable' (i.e., in $L^p(\Omega)$), and gives us a bound on the $L^p$-norm in terms of the $L^q$-norm.

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 of Theorem B.40 (Sobolev)

The theorem is known as the Sobolev embedding theorem, which is a central result in the theory of Sobolev spaces. Sobolev spaces are a type of function space equipped with a norm that measures both the size of a function and its derivatives. They play a fundamental role in the theory of partial differential equations and the calculus of variations.

Here is a sketch of the proof in the case $d > p > 1$:

We first note that the space $W^{1,p}(\mathbb{R}^d)$ consists of functions that are in $L^p(\mathbb{R}^d)$ and whose weak derivatives are also in $L^p(\mathbb{R}^d)$.

Let’s denote by $B$ the unit ball in $L^p(\mathbb{R}^d)$ with respect to the $L^p$-norm of the gradient. We can show that $B$ is precompact in $L^{p^*}(\mathbb{R}^d)$ with respect to the weak topology. This is known as the Rellich-Kondrachov theorem.

Therefore, for any $u \in W^{1,p}(\mathbb{R}^d)$, we can find a sequence $u_n \in B$ such that $u_n \rightharpoonup u$ in $L^p(\mathbb{R}^d)$ (weak convergence), and $u_n \to u$ in $L^{p^*}(\mathbb{R}^d)$ (strong convergence).

Since $u_n \in B$, we have $\|\nabla u_n\|_{L^p(\mathbb{R}^d)} \leq 1$. Therefore, by the Poincaré inequality, we have

$\|u_n\|_{L^{p^*}(\mathbb{R}^d)} \leq C \|\nabla u_n\|_{L^p(\mathbb{R}^d)} \leq C,$

for some constant $C$ independent of $n$.

Taking the limit as $n \to \infty$, we obtain

$\|u\|_{L^{p^*}(\mathbb{R}^d)} \leq C \|\nabla u\|_{L^p(\mathbb{R}^d)}.$

This proves the theorem.

 This proof is a sketch and omits many technical details. The complete proof requires a deep understanding of functional analysis and measure theory, and involves several advanced results such as the Poincaré inequality and the Rellich-Kondrachov theorem. See for more details [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 of Corollary B.41

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

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

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

## 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 of Lemma B.61 (Poincaré)

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 (Sobolev). 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 of Lemma B.63

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) $\square$