Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
You can do it in a $C^2$ setting by Perron's method, for example. –  timur Sep 6 '12 at 23:31
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
add comment

1 Answer

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.

share|improve this answer
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
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.