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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?

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1 Answer 1

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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.

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  • $\begingroup$ Thank you very much. Could you name the title of the Li-Tam's paper. $\endgroup$
    – DLIN
    Jan 14, 2019 at 3:04
  • $\begingroup$ Symmetric Green's Functions on Complete Manifolds. $\endgroup$
    – EveryLT
    Jan 14, 2019 at 4:48
  • $\begingroup$ Thank you very much for the replying. Might I ask a further Question: If $f$ belongs $L^{p}_k$, can we say that $u\in L^p_{k+2}$ for $k\geq0$. Namely, the regularity for $\Delta$ $\endgroup$
    – DLIN
    Jun 13, 2019 at 12:21
  • $\begingroup$ Not an expert, just happened to know Malgrange's result. I think you may look at Hormander's book or `Differentiable Manifold' by de Rahm. Certain regularity is discussed, but maybe not in term of Sobolev spaces. $\endgroup$
    – EveryLT
    Jun 14, 2019 at 13:32

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