Let $U=\{u\le0\}$ and $\partial U=\{u=0\}$. Suppose $\nabla u$ does not vanish on $\partial U$. Then the (canonical extension of the) normal vector field to $\partial U$ (pointing to the interior of $\mathbb R^n \setminus U$ is given by $$\nu = \frac{\nabla u}{|\nabla u|}.$$
This formula is very helpful in some elliptic problems because it is so related to $\int |\nabla u^2| = -\int u \Delta u.$
What about elliptic problems with the fractional Laplacian $(-\Delta)^s$ ($s \in (0,1)$)?
- Is the existence of a solution in $H^{s}$ or $H^{2s}$ even enough to make sense of the normal vector above?
- Can $\int |(-\Delta)^{s/2}u|^2 = \int u (-\Delta)^s u$ be used to control $\nu$?
- Or do people use a different notion of normal vector?