Best constant for Sobolev inequality from Sobolev space to Hölder space

Question

Let $u(x) \in W^{1,p}(\mathbb{R}^n)$ such that the Sobolev embedding from $W^{1,p}(U)$ to $C^{0,\alpha}(U)$ holds for some $n$, $p$, $\alpha$ and bounded domain $U$. We have the Sobolev inequality $$ \|u\|_{C^{0,\alpha}(U)} \leq C \|u\|_{W^{1,p}(U)}. $$ I am wondering how the constant $C$ depends on the domain $U$. Suppose $u$ is bounded and take maximum value at the origin. It is obvious that $\|u\|_{W^{1,p}(B_1)}\leq \|u\|_{W^{1,p}(B_4)}$ and $$ \|u\|_{C^{0}(B_1)} \leq C_1 \|u\|_{W^{1,p}(B_1)}, \quad \|u\|_{C^{0}(B_1)}=\|u\|_{C^{0}(B_4)} \leq C_4 \|u\|_{W^{1,p}(B_4)}.$$ So I expect that $C_1$ should be larger than $C_4$ as the LHSs are the same, but $\|u\|_{W^{1,p}(B_4)}$ on the RHS of the second seems larger. Is there any explicit relation of $C$ on the domain?