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Exercise :

Let $\Omega \subseteq \mathbb R^n$ be open and bounded, $\{u_n, u\}_{n \geq 1} \subseteq L^p(\Omega)$ with $1<p<\infty$ and we assume that $\|u_n\|_p \to \|u\|_p, \; u_n \xrightarrow{a.e.} u$. Show that $u_n \to u$ in $L^p(\Omega)$.

Attempt :

Since $\|u_n\|_p \to \|u\|_p$ then it is :

$$\|u_n\|_p^p \to \|u\|_p^p \Leftrightarrow \int_\Omega |u_n|^p \mathrm{d}x = \int_\Omega |u|^p\mathrm{d}x$$

To show that $u_n \to u$ in $L^p(\Omega)$, we need to show that $\|u_n - u\|_p \to 0$. But :

\begin{align*} \|u_n-u\|_p^p &= \int_\Omega |u_n-u|^p\mathrm{d}x \\ &\leq \int_\Omega \left(|u_n| + |u|\right)^p\mathrm{d}x \end{align*}

But from then on, I don't see how to continue in order to derive an expression of the form $\|u_n-u\|_p^p < \varepsilon$ where $\varepsilon > 0$.

Any hints will be greatly appreciated.

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Since we know that $u_n\to u$ almost everywhere we can conclude that $|u_n-u|^p\to 0$ almost everywhere. Hence: (we will use Fatou lemma)

$2\int_{\Omega} |u|^pdx=\int_{\Omega}\lim_{n\to\infty}(|u|^p+|u_n|^p-|u-u_n|^p)dx\leq$

$\leq\liminf_{n\to\infty}\int_{\Omega}(|u|^p+|u_n|^p-|u-u_n|^p) dx=2\int_{\Omega} |u|dx-\limsup_{n\to\infty}\int_{\Omega} |u-u_n|^p dx$

In the last equality we used the fact that $\int_{\Omega} |u_n|^p\to\int_{\Omega} |u|^p$.

So what we got is that $\limsup_{n\to\infty}\int_{\Omega} |u-u_n|^pdx\leq 0$. Since this is a sequence of non negative real numbers this of course implies $\lim_{n\to\infty}\int_{\Omega} |u-u_n|^pdx=0$.

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