1
$\begingroup$

Let $(M,g)$ be a complete non-compact manifold with bounded geometry, such that the Sobolev embeddings hold and $C^\infty_c$-functions are dense in $L^p_k$ space.

For the equation $$\Delta u=f,$$ for some $f\in L^2(M)$.

Q How can we find a solution $u\in L^2$ satisfies the above equation?

Is there any related work?

$\endgroup$
4
$\begingroup$

In fact, for any noncompact Riemannian manifold $M$ (not necessarily with bounded geometry) and for any distribution $f$, there is a distribution $u$ such that $Du=f$, where $D$ is an elliptic operator satisfying the unique continuation property.

This was proved by Malgrange in Existence et approximation des solutions des ́equations aux d ́eriv ́ees partielles et des ́equations de convolution'. The Euclidean case was discussed in Hormander's bookLinear partial differential operators' (Chapter 4 as I remembered), and the arguments therein carries over to general Riemannian manifolds.

There is also a constructive proof due to Li and Tam, which is easier to read for geometers.

$\endgroup$
  • $\begingroup$ Thank you very much. Could you name the title of the Li-Tam's paper. $\endgroup$ – DLIN Jan 14 at 3:04
  • $\begingroup$ Symmetric Green's Functions on Complete Manifolds. $\endgroup$ – EveryLT Jan 14 at 4:48

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

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