# normal derivate boundary

Let $$f:\mathbb{R}^{n}\to\mathbb{R}$$ with $$u=0$$ in $$\mathbb{R}^{n}\backslash \overline{D}$$, where $$D$$ is a bounded domain and consider $$d$$ a smooth version of the distance to the boundary of $$\partial D\in C^{\infty}$$. Determine the normal derivative $$\partial\nu (f/d^{\alpha-1})|_{\partial\Omega}$$ with $$\alpha>0$$.

My approach: Note that, $$\partial_{\nu}(\frac{f}{d^{\alpha-1}})|_{\partial\Omega}=$$

$$\partial_{\nu}((fd^{-\alpha})d)|_{\partial\Omega}=[d\nabla(fd^{-\alpha})+fd^{-\alpha}\nabla d]|_{\partial\Omega}=-fd^{-\alpha}|_{\partial\Omega}\nu$$

where $$\nu$$ is the outward unit normal to $$\partial\Omega$$.

But in case $$\alpha=1$$, I feel that this definition is not correct with what I would expect in the usual case. Any way, also I can see the following, $$\partial_{\nu}(\frac{f}{d^{\alpha-1}})|_{\partial\Omega}=[d^{1-\alpha}\nabla f + f(1-\alpha)d^{-\alpha}\nabla d]\cdot\nu|_{\partial\Omega}$$ Therefore, I feel that the notion of normal derivative on the boundary of $$\partial\Omega$$ is not well defined, can someone help me deal with these terms, greetings