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
Proof. This is a consequence of Hölder’s inequality.
One of the key arguments in the embedding theory is the following:
Proof. See [MaZ97, p. 32], [Sob63, §I.7.4], or [Bre91, p. 162].
Proof. See [MaZ97, p. 34], [Sob63, §I.8.2], or [Bre91, p. 165].
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
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).
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) \(\square\)