I will not be rigirous cause my question does not require this. It is well known how to treat the solution (for example) of the problem $\Delta u=f$ in $\Omega$ and $u=g$ on $\partial \Omega$ by the introduction of some Sobolev Space. My quite vague question is the following one: Is it possible to give a meaning to the previous problem and to prove the existence of at least a solution of the previous problem in the space of distributions without introducing Sobolev Spaces?

Here's a rough approach which I do not believe needs to use Sobolev space theory, though it's perhaps not what you are looking for: Assuming some boundary regularity (I believe that $\partial \Omega \in C^2$ is sufficient), one can merely recast the problem as solving for a Green's function. It is then possible to prove the existence of the Green's function for Laplace's equation using somewhat elementary means, if I recall. 


I think what you are looking for might be explained in Section 18.3 and Appendix B.2 in Lars Hörmander, The Analysis of Linear Partial Differential Operators III, SpringerVerlag. I do not understand enough, but Hörmander says "Section 18.3 is intended to give a solid framework for the study of boundary problems, in particular a discussion of spaces of distributions in a manifold with boundary and their wave front sets. However, little use will be made of the results in the later chapters where we use more conventional but noninvariant techniques. The notions introduced in Section 18.3 seem so natural though that they are bound to play an increasingly important role." Maybe somebody could comment, what has happened since (the book dates from 1985) with this theory. 

