Let $\Omega$ be a bounded region in $R^n$ and define $W:=\{ u \in H^{1}(\Omega): u(x_0)=0 \},$ where $x_0 \in \partial \Omega$ is a fixed point. Is there a constant $C$ such that
$||u||_{L^2(\Omega)} \leq C ||\nabla u||_{L^2(\Omega)}$,
for all $u\in W$?
This inequality cannot hold. Here is a counterexample:
Let $\phi_{\epsilon}$ be a bump function supported on $B_{\epsilon}(0) \subset \mathbb{R}^n$ with value $1$ at $0$ and satisfying $|\nabla \phi_{\epsilon}| < \frac{2}{\epsilon}$. Let $u_{\epsilon} = 1-\phi(x-e_n)$, the function which is 1 everywhere but dips quickly to $0$ at $e_n$. Then $$\int_{B_1}u_{\epsilon}^2 dx$$ is like $1$ but $$\int_{B_1} |\nabla u_{\epsilon}|^2 \leq C\epsilon^{n-2},$$ which goes to $0$ for $n \geq 3$.
Addendum: In dimension $1$ the inequality holds by fundamental theorem. Dimension $2$ is a funny critical case that barely doesn't work; Take $\phi(x) = \log\log(1+1/|x|)$ and let $\phi_r(x) = \log\log(1+1/r) - \phi(x)$. In $B_1((1+r)e_1)$ we have $\int \phi^2$ going like $(\log\log(1+1/r))^2$ which goes to $\infty$ slowly as $r \rightarrow 0,$ but computing $|\nabla \phi_r|^2$ we get something going like $$\frac{1}{|x|^2|\log(1+1/|x|)|^2},$$ which is integrable near $0$.
