I was trying to think about the relationship between Stoke's Theorem and Distribution Theory when I realized that if we think about the function space $L_2(\mathbb{R}^n) \otimes Ext(\mathbb{R}^n)$ (space of alternating forms in $R^n$ with inner product: $$\langle u, v\rangle = \int_{R^n} *((*u)\wedge v)$$
where $*$ is the hodge star operator. Then Stokes theorem is simply stating that $$\langle du, \Omega\rangle = \langle u, \delta \Omega \rangle$$ where $\Omega$ is the $n$-form which serves as an indicator function for an $R^n$ dimensional manifold, and $u$ is a $(n-1)$ - form differential form. ($d$ is the exterior derivative and $\delta$ it's adjoint)
Now I'm just wondering can boundary conditions and other aspects of PDEs be thought about in a similar way? (where we think of the boundary as $\delta\Omega$)
Also are there any references for something similar to this?