# 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

## 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:

## 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

for some constant \(C\) independent of \(n\).

Taking the limit as \(n \to \infty\), we obtain

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]. |

## Proof of Corollary B.41

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

## Proof of Theorem B.46 (Rellich-Kondrachov)

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

## 2. Poincaré-like inequalities

## 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).

## 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) \(\square\)