In the theory of $\mathbb{R}^{d}$, one defines the space of Schwartz functions $ \mathcal{S}(\mathbb{R}^{d}) $ to be the space of smooth functions decaying faster than any polynomials. However, when one consider bounded domains $\mathit{\Omega}$, the natural adaptation to this definition would be $ \mathcal{D}(\mathit{\Omega}) $ which is the space of smooth functions with compact support in $\mathit{\Omega}$.
When dealing with boundary value problems for partial differential equations, one looks for solutions in Sobolev spaces $H^{s}(\mathit{\Omega})$ or $H^{s}_{0}(\mathit{\Omega})$. In most literatures one defines these space directly by density, whereas in the case of $\mathbb{\mathbb{R}^{d}}$ I know how $H^{s}(\mathbb{R}^{d})$ can be defined from distributions, that is
$$H^{s}(\mathbb{R}^{d}) = \Big\{ u \in \mathcal{S}'(\mathbb{R}^{d}) \ \Big/ \ \hat{u} \in L^{2}_{loc}(\mathbb{R}^{d}), \ \int_{\mathbb{R}^{d}} ( 1 + |\xi|^{2} )^s |\hat{u}(\xi)|^2 d\xi < \infty \Big\}$$ Here the hat denotes the Fourier transform $\mathcal{F}: \mathcal{S}'(\mathbb{R}^{d}) \rightarrow \mathcal{S}'(\mathbb{R}^{d}).$ It is natural to guess that, we should have the definitions $$H^{s}_{0}(\mathit{\Omega}) = \Big\{ u \in \mathcal{D}'(\mathit{\Omega}) \ \Big/ \ \hat{u} \in L^{2}_{loc}(\mathit{\Omega}), \ \int_{\mathit{\Omega}} (1+ |\xi|^2)^s |\hat{u}(\xi)|^2 d\xi < \infty \Big\}$$ and this is alright. But what about the case of $H^{s}(\mathit{\Omega})$? Is it $$ H^{s}(\mathit{\Omega}) = \Big\{ u \in \mathcal{S}'(\mathbb{R}^{d}) \ / \ \hat{u} \in L^{2}_{loc}(\mathcal{\mathit{\Omega}}), \ \int_{\mathit{\Omega}} (1+ |\xi|^2)^{s} |\hat{u}(\xi)|^2 d\xi < \infty \Big\} ? $$ The distinction is that, in order to make sense of the boundary one must look at the trace map $ \gamma: H^{s}(\Omega) \rightarrow H^{s-1/2}(\partial \Omega) $ since the boundary is not already made sense of. Coming directly from the spaces $\mathcal{S}'(\mathbb{R}^{d})$ or $\mathcal{D}'(\mathit{\Omega})$, non of the differential operators on these spaces respect the boundary $\partial \mathit{\Omega}$. In particular, I feel like for open bounded domain the differential operator should incorporate a boundary term from intergatin by part, something like $$ \langle \partial_{x} u, \phi \rangle = - \langle u, \partial_{x} \phi \rangle + boundary\ term \ which \ make \ sense \ somethow $$ just to get a feel what I am not sure about.
This suggests that either
Something is wrong with my definition.
The standard is to treat the boundary separately using the trace map $\gamma$.
This thus gets me ask the following questions:
Is my guess on the definition and guesses on treating the boundary value of distributions correct?
If so, is there a natural way to make sense of $\gamma$ in terms of distributions? Is it $\langle \gamma u, \phi \rangle = \langle u, \gamma \phi \rangle$ where $$ \gamma : \mathcal{S}(\mathbb{R}^{d}) \rightarrow \mathcal{S}({\mathbb{R}^{d}}): \gamma \phi = \phi_{|_{\partial \Omega}} ? $$
I find it hard to gather these information from the literature. Hence the question.
I would appreciate any insight! Many thanks in advance!