# For which functions $u$ is $\partial_\nu u \in H^{-1/ 2}(\partial\Omega)$?

Let $\Omega$ be a smooth bounded domain, with outward normal vector $\nu$. Under what conditions on a function $u$ do we get that

$$\partial_\nu u \in H^{-1/ 2}(\partial\Omega).$$

That is, under what conditions is the trace of the normal derivative of $u$ in the dual fractional Sobolev space on the boundary?

I know that $u \in H^2$ suffices, but is there anything weaker?

• How do you define $\partial_\nu u$? – gerw Jan 25 '18 at 10:16