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I am studying a proof, and the author imposes a condition to a function. It seems strange to me since I do not see any non-null function satisfying this condition.

Let $u\in C^2(\mathbb R\times (0,+\infty))$ satisfying

$$\sup_{y\geqslant 0} \int_{\mathbb R} \vert u(x,y)\vert \mathrm d x=0.$$

Does there exist such a function $u$ which is not null everywhere?

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$u$ would have to be $0$ everywhere. This supremum being zero means that for every $y$ $\int_{\mathbb{R}} |u(x,y)|dx=0$. Which you can use to show (by continuity of $u$) that for any fixed $y$, $u(x,y)=0$ for all $x$. But this means that for any pair $(x,y)$, that $u(x,y)=0$.

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    That is what I thought... I do not get this silly condition then. – 2017-02-26
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    My guess, without seeing the context, is that it is supposed to be lim sup, not sup. – 2017-02-26
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    I asked another question to add context if you want to take a look :) http://math.stackexchange.com/questions/2162078/sup-limsup-or-else-what-is-the-error-here – 2017-02-26