Is there a Log-Sobolev inequality for Lebesgue measure on $[0,1]$ on compact subsets of $\mathbb R^n$?

After searching on the internet for long enough, I would like to pose the question here. I hope there is no duplicate (if there is please let me know)

Is it true that, there is a universal constant $C>0$ satisfying

$$ \int^1_0 f^2 \log f^2 dx \leq C \int_0^1|f'|^2dx$$

for all $f$ weakly differentiable such that $\int_0^1 f^2(x) dx = 1$, where $dx$ is the Lebesgue measure on $[0,1]$?

If this is true, can it be extended to the Lebesgue measure on any compact sets of $\mathbb R^n$? That is, let $K$ be a compact set of $\mathbb R^n$ (put some more assumptions if you like), is there a universal constant $C = C(n,K)>0$ satisfying

$$ \int_K f^2 \log f^2 dx \leq C \int_K|f'|^2dx $$

for all $f$ weakly differentiable and such that $\int_K f^2(x) dx = 1$?