Let $\Omega$ denote a bounded smooth domain in $R^N$ and consider $\Gamma$ a smooth subset (assume its some $k$ dimensional manifold where $k \le N-1$). Let $ \delta(x)=dist(x, \Gamma)$. On occasion you might have some pde that holds on $\Omega \backslash \Gamma$ and you want to prove by a density argument some result.
For instance suppose we have $ \Delta w(x)= 0 $ in $ \Omega \backslash \Gamma$ with $ w=0$ on $ \partial \Omega$ and we have $ w$ is bounded and we would like to show that $ w=0$. If $dim(\Gamma)<N-2$ then we can prove this by multiplying the pde by $ \gamma_{\epsilon}^2 w(x)$ where $ \gamma_\epsilon$ is a cut-off which is zero near $\Gamma$. We can use
$$ \gamma_\epsilon(x)= \frac{\delta(x)-\epsilon}{\epsilon}$$ for $ \{x: \epsilon <\delta(x)<2 \epsilon\}$ and extend in the obvious way outside this set.
If $ dim(\Gamma)=N-2$ then this method fails. I recall there is a smarter way to pick the cut off (that might involve two parameters and a log) that allows one to still handle this higher dimensional case. Any comments would be appreciated. thanks
