Ngô Quốc Anh

February 28, 2012

The Rellich embedding theorem on a bounded domain

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 18:18

Today, let us summarize steps during the standard proof of the Rellich embedding theorem on a bounded domain \Omega \subset \mathbb R^n. This theorem says that

Theorem (Rellich). Suppose that \Omega \subset \mathbb R^n is an open, bounded domain with C^1 boundary,  and that 1<p< n. Then W^{1,p}(\Omega) is compactly embedded in L^q(\Omega) for all 1\leqslant q <\frac{np}{n-p}.

In particular, for any sequence \{u_j\}_j \subset W^{1,p}(\Omega) , there exists a subsequence \{u_{j_k}\}_k \subset \{u_j\}_j such that u_{j_k} \to u strongly in L^q(\Omega) for some u \in L^q(\Omega).

In order to prove the Rellich theorem, we need the so-called Arzela-Ascoli theorem.

Lemma (The Arzela-Ascoli theorem). Suppose that u_j \in C^0(\overline\Omega), \|u_j\|_{C^0(\overline\Omega)} \leqslant M < \infty, and \{u_j\}_j is equicontinuous. Then there exists a subsequence u_{j_k} \to u uniformly on \Omega.

The Arzela-Ascoli theorem is well-known. To prove the Rellich theorem, we shall use the standard mollifiers. To do that, we have to extend \Omega a little bit.

Step 1. Assume that \overline\Omega \subset \mathbb R^n is also an open, bounded domain with C^1 boundary with \Omega \subset\subset \overline \Omega. By the Sobolev extension theorem, we can define Eu_j by \overline u_j with

\sup_j \|\overline u_j\|_{W^{1,p}(\mathbb R^n)} < C<\infty.

By the Gagliardo-Nirenberg inequality,

\sup_j \|\overline u_j\|_{L^q(\Omega} < C<\infty.

For \varepsilon>0, let \eta_\varepsilon denote the standard mollifiers and set \overline u_j^\varepsilon = \eta_\varepsilon * \overline u_j. By choosing \varepsilon sufficiently small, \overline u_j^\varepsilon \in C^\infty(\overline \Omega).

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May 5, 2010

An extension of the Rellich-Kondrachov theorem

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 14:00

In mathematics, the Rellich-Kondrachov theorem is a compact embedding theorem concerning Sobolev spaces. It is named after the Italian-Austrian mathematician Franz Rellich.

Theorem (Rellich-Kondrachov). Let \Omega \subset \mathbb R^n be an open, bounded Lipschitz domain, and let 1 \leqslant p \leqslant kn. Set

\displaystyle p^\star := \frac{np}{n - kp}.

Then the Sobolev space W^{k,p}(\Omega) is continuously embedded in the L^q(\Omega) space for every 1 \leqslant q \leqslant p^\star and is compactly embedded in L^q(\Omega) for every 1 \leqslant q < p^\star. In symbols,

\displaystyle W^{k, p} (\Omega) \hookrightarrow L^{p^\star} (\Omega)

and

\displaystyle W^{k, p} (\Omega) \subset \subset L^{q} (\Omega) for 1 \leqslant q < p^\star.

It is worth noticing from the theory of Sobolev spaces that

\displaystyle W^{0, p} (\Omega) \equiv L^p(\Omega).

Therefore, we have the following extension

Theorem (Extension of Rellich-Kondrachov). Let \Omega  \subset \mathbb R^n be an open, bounded Lipschitz domain, and let 1 \leqslant p \leqslant kn. Set

\displaystyle p^\star := \frac{np}{n - kp}.

Then we have

\displaystyle W^{j+k, p} (\Omega) \hookrightarrow W^{j, q} (\Omega) for 1 \leqslant q \leqslant p^\star

and

\displaystyle W^{j+k, p} (\Omega) \subset \subset W^{j,q} (\Omega) for 1 \leqslant q < p^\star.

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