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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?

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You can do it in a $C^2$ setting by Perron's method, for example. –  timur Sep 6 '12 at 23:31
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I believe what you want is carried out in Chapter 5 of the book "Introduction to the Theory of Linear Partial Differential Equations" by Chazarain and Piriou. You can view some if not all of the book in Google Books. –  Deane Yang Sep 8 '12 at 21:15

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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.

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Along the same lines, you can take convolution of the fundamental solution or Green function and a distribution, that is $f$, and that will give the solution. –  S.A.A Nov 10 '12 at 5:51

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