Functions with finite Dirichlet integral in a domain with a cusp at the boundary

Let $\Omega$, be a bounded domain in $\mathbb R^2$, $n>2$, with inward or outward cusps at $\partial\Omega$, and let $H^1(\Omega)$ be the space of functions with finite Dirichlet integral. Our main result is a characterization of the space of traces of functions in $H^1(\Omega)$. As a corollary we obtain the existence of a continuous linear extension mapping: $H^1(\Omega)\to H^1(\mathbb R^n)$ provided the domain has an inward cusp. (It is well known that the latter fails for $n=2$).