I am looking for the optimal embedding result $H_0^2(0,1)\hookrightarrow L_{\Phi}(0,1)$. This includes finding the largest possible Yong function $\Phi(x)$ (the smallest space $L_\Phi(0,1)$) for which the embedding holds.
The classical Sobolev embedding result tells us that $H_0^2(0,1)\hookrightarrow C^{1,1/2}(0,1)$. Also, a Trudinger's embedding result ensures that $H^1_0(0,1)\hookrightarrow L_{\Psi}(0,1)$ where $\Psi(0,1)=e^{t^2}-1$.
Since we have second order derivatives, it seems reasonable to allow the function $\Phi(x)$ grows faster than $\Psi(x)$, i.e. $$\lim_{x\to \infty} \frac{\Phi(x)}{\Psi(x)} =\infty$$
but how much faster is a question here?