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) \(\square\)