I have only seen the following version of 2D [Ladyzhenskaya inequality][1] in cited references of PDE: 

>Let $\Omega$ be a Lipschitz domain in ${\bf R}^2$ and let $u: \Omega → {\bf R}$ be a weakly differentiable function that vanishes on the boundary of ${\bf R}$ in the sense of trace (that is, $u$ is a limit in the Sobolev space $H^1(\Omega)$ of a sequence of smooth functions that are compactly supported in $\Omega$). Then there exists a constant $C$ depending only on $\Omega$ such that
$$
{\displaystyle \|u\|_{L^{4}}\leq C\|u\|_{L^{2}}^{1/2}\|\nabla u\|_{L^{2}}^{1/2}}. 
$$

Is it true for periodic functions as well? More precisely, is it true that
$$
{\displaystyle \|u\|_{L^{4}(\Omega)}\leq C\|u\|_{L^{2}(\Omega)}^{1/2}\|\nabla u\|_{L^{2}(\Omega)}^{1/2}} 
$$
where $u:{\bf R}^2\to{\bf R}$ is a smooth function with the period $\Omega=[l_1,r_1]\times[l_2,r_2]$?


  [1]: https://en.wikipedia.org/wiki/Ladyzhenskaya%27s_inequality