I have a problem with this exercise (see the text in the following link).
Interpolation like inequality ,Question from Brezis' book exercise 8.15
The link practically solves it. Only one last step is missing, namely proving the inequality when $u(0)\neq 0$. Brezis suggests using $\zeta u$. I tried to do the calculations, but there is something that doesn't add up for me. Since $\zeta(t)=1$ for $t\in[1/2,1]$, for this interval, I have the desired inequality. But how do I handle $[0,1/2)$?