MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Consider 1 < $p<\infty$ and an integer $k$. Does interior elliptic regularity for the Laplacian also hold in the Sobolev space $W^{k,p}$ of negative order?

More precisely I am interested in the following question: Let $u\in W^{-1,p}(R^n)$ be a distributional solution of $\Delta u=Su,$ where $S$ is smooth. Is it then true that $u$ is smooth?

share|cite|improve this question
I'd be very surprised, if this is not discussed in your favorite reference for distributions and Sobolev spaces of negative order. – Deane Yang Oct 25 '11 at 19:34
You might also want to post this question on It's not really suitable for MO. – Deane Yang Oct 25 '11 at 19:35
It is true that a distribution T whose distributional laplacian is zero, $\Delta T = 0$, is actually $T = T_f$ for some smooth harmonic function $f$. What is S, sorry? – Spencer Oct 25 '11 at 19:40
I posted it on, but did not get a useful answer. So I decided to try it here. – Orbicular Oct 25 '11 at 20:28
@Spencer: S is a smooth function. – Orbicular Oct 25 '11 at 20:33
up vote 2 down vote accepted

The smoothness result holds even for solutions from $\mathcal D'(\mathbb R^n)$. See, for example, Theorem IX.26 in Vol.2 of "Methods of Modern Mathematical Physics" by Reed and Simon.

share|cite|improve this answer

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.