approach presented in [@Triebel78].

Let $D$ be a sufficiently smooth domain and $s\in{{\mathbb R}}$. We denote by $H^s(D)$ the Sobolev space of order $s$ on the set $D$ which we define as a restriction (in the sense of distributions) of the space $H^s({{\mathbb R}}^d)$ (introduced via Fourier transform). We define the norm in $H^s(D)$ by the relation $$\|u\|_{s,D}^2=\inf\left\{\|w\|_{s,{{\mathbb R}}^d}^2\, :\; w\in H^s({{\mathbb R}}^d),~~ w=u ~~
\mbox{on}~~ D
    \right\}$$ We also use the notation $\|\cdot \|_{D}=\|\cdot \|_{0,D}$ and $(\cdot,\cdot)_D$ for the corresponding $L_2$ norm and inner product. We denote by $H^s_0(D)$ the closure of $C_0^\infty(D)$ in $H^s(D)$ ([[with respect to ]{}]{}$\|\cdot \|_{s,D}$) and introduce the spaces $$H^s_*(D):=\left\{ f\big|_D\, :\; f\in H^s({{\mathbb R}}^d),\;
{\rm supp }\, f\subset \overline{D}\right\},\quad s\in {{\mathbb R}}.$$ Below we need them to describe boundary traces on ${{\Omega}}\subset\partial {{\cal O}}$. We endow the classes $H^s_*(D)$ with the induced norms $\|f \|^*_{s,D}= \| f \|_{s,{{\mathbb R}}^d}$ for $f\in H^s_*(D)$. It is clear that $$\|f \|_{s,D}\le \|f \|^*_{s,D}, ~~ f\in H^s_*(D).$$ However, in general the norms $\|\cdot \|_{s,D}$ and $\|\cdot \|^*_{s,D}$ are not equivalent. It is known that (see [@Triebel78 Theorem 4.3.2/1]) that $C_0^\infty(D)$ is dense in $H^s_*(D)$ and $$\begin{aligned}
& H^s_*(D)\subset H^s_0(D)\subset H^s(D),~~~ s\in{{\mathbb R}};\\
& H^s_0(D) =  H^s(D),~~~ -\infty< 