I think the most natural assumption is that $D$ is a bilipschitzian image of a smooth domain $D_0$, since a change of variables $A:D_0\to D$ with $a|x-y|\le|A(x)-A(y)|\le b|x-y|$ ($a>0$) preserves $H^1(D_0)$ and the space of traces $H^{1/2}(\partial D_0)$ as defined with $\int\int\frac{|f(x)-f(y)|^2}{|x-y|^{d+1}}\ dx\ dy<\infty$.

I think the most natural assumption is that $D$ is a bilipschitzian image of a smooth domain $D_0$, since a change of variables $A:D_0\to D$ with $a|x-y|\le|A(x)-A(y)|\le b|x-y|$ ($a>0$) preserves $H^1(D_0)$ and the space of traces $H^{1/2}(\partial D_0)$ as defined with $\int\int\frac{|f(x)-f(y)|^2}{|x-y|^d}\ d\sigma(x)\ d\sigma(y)<\infty$.