p-Dirichlet spaces over chord-arc domains

Let $Γ$ be a rectifiable Jordan curve in the complex plane, let $Ω_i$ and $Ω_e$ be its interior and exterior domains, respectively, and let $1 < p < \infty$. Let $E$ be the vector space of restrictions to $Γ$ of functions in $C^1(\mathbb C)$. We consider the following three seminorms on $E$: (i) $\l