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Suppose that $f,u,u_n\in L^p(\Omega)$ where $\Omega\subset\mathbb{R}^N$ is a bounded domain and $f,u,u_n\geq 0$. Suppose $\|u+u_n\|_p\rightarrow \|u+f\|_p,$ $u_n\rightarrow f\text{ a.e. in } \Omega,$ and $u_n>f.$

Does this implies that $\|u_n-f\|_p\rightarrow0\,?$

Thanks.

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    yes its @richard2012-11-12

1 Answers 1

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If $\|u\|_p=\infty$, it is not necessarily true. For example, $N=1$, $\Omega=[0,1]$, $u_n=n\cdot\mathbf{1}_{[0,\frac{1}{n}]}$ and $f=0$.

If $\|u\|_p<\infty$, it it true without assuming that $u_n>f$. The proof is as follows. Let $g_n=2^{p-1}(|u+u_n|^p+|u+f|^p)-|u_n-f|^p\ge 0,$ by Fatou's lemma and your assumptions,

$2^p\|u+f\|_p^p=\int_\Omega \liminf_{n\to\infty} ~g_n\le \liminf_{n\to\infty}\int_\Omega g_n=2^p\|u+f\|_p^p-\limsup_{n\to\infty}\int_\Omega |u_n-f|^p.$

The conclusion follows.

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    Yes you are right, thank you @ri$c$hard2012-11-13