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 $u\in W$?



Sobolev Inequality
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 $0$.


