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) \(\square\)