# 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. 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, $\overline u_j^\varepsilon \in C^\infty(\overline \Omega)$.

## 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 $1 \leqslant q < p^\star$.