In a paper of Brezis and Gallouet I don't understand a step:
For $\phi\in C_0^\infty(\mathbb{R}^2)$ we have this inequality:
(1) $\quad$ $\int_\mathbb{R^2}|\phi|^2\leq \frac14\int_\mathbb{R^2}|\phi_{x_1}|dx\int_\mathbb{R^2}|\phi_{x_2}|dx$
Where $\phi_{x_1}$ denotes $\frac{\partial \phi}{\partial x_1}$.
Choosing $\phi=|u|^2$ leads to:
(2)$\quad \int |u|^4dx\leq\int |u|^2dx(\int |u_{x_1}|^2dx)^\frac12(\int |u_{x_2}|^2dx)^\frac12$
Thus:
(3)$\quad \int |u|^4dx\leq \frac12 \int |u|^2dx\int |\nabla u|^2dx$
I suppose that we use Cauchy-Schwartz to obtain (2) but I don't understand how to obtain (3).
Thank you for your answers !
The second inequality is indeed by Cauchy-Schwarz and using $(1)$ with the particular choice of $\phi$.
For the third inequality, just begin with the second, then use the elementary inequalities : $2ab\leq a^2+b^2$ and $(\partial_1 u)^2\leq |\nabla u|^2 = (\partial u)^2+(\partial u)^2.$