a) Let $a<b$. Prove there exists an universal contant $C>0$ such that $$ \|u\|_{L^{\infty}(a, b)} \leq C\|u\|_{H^{1}(a, b)} \quad \forall u \in H^{1}(a, b) $$
b) Let $a<b$. Prove that the injection $H^{1}(a, b) \rightarrow L^{\infty}(a, b)$ is compact.
c) Let $\Omega \subset \mathbb{R}^{2}$ a bounded open set with boundary of class $C^{1} .$ Prove that there no exists a constant $C>0$ such that $$ \|u\|_{L^{\infty}(\Omega)} \leq C\|u\|_{H^{1}(\Omega)} \quad \forall u \in H^{1}(\Omega) $$
So I proved the part a) indeed the constant is independet from $p$ in this case $p=2.$ For b) I have no idea because I saw the same problem with the compact $[a,b]$ but for that problem the solution used the Arzelá-Ascoli Theorem so compactness is important, so I have no idea how to prove b). And c) is also a mystery.
You said you can do b) with compact [a,b]. This approach can be salvaged because you can extend an $H^1$ function from $(a,b)$ to $[a,b]$ as a (Hölder) continuous function.
For c) try to construct a radially symmetric singular function whose $H^1$ norm is finite.