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I'm currently reading the paper on the Yamabe problem by Lee and Parker, and am looking for a reference for Theorem 2.8.

Theorem 2.8 (Existence of the Green Function).

Suppose $M$ is a compact Riemannian manifold of dimension $n\ge 3$, and $h$ is a strictly positive smooth function on $M$. For each $P\in M$, there exists a unique smooth function $\Gamma_P$ on $M\setminus \{P\}$, called the Green function for $\Delta + h$ at $P$, such that $(\Delta + h)\Gamma_P = \delta_P$ in the distribution sense, where $\delta_P$ is the Dirac measure on $P$.

Does anyone know where I might find out about general existence results for Green's functions (in particular this one)? Thanks!

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Chapter 4 of Aubin's Nonlinear Analysis on Manifolds constructs the Green's function for the Laplacian on a Riemannian manifold (without the $h$). –  Henry T. Horton Feb 17 '13 at 19:57
    
@HenryT.Horton: Thanks! I'll check it out. –  Sam Feb 17 '13 at 23:13

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up vote 4 down vote accepted

Sorry we weren't more explicit about giving references. I haven't come up with a good reference for an explicit proof of our Theorem 2.8; but here are three suggestions for constructing your own proof.

  1. Following Henry Horton's suggestion, you can adapt the proof in Aubin's Nonlinear Analysis on Manifolds to this case.
  2. As Tom Parker and I commented after the proof of Theorem 6.5 in our paper, that proof can be adapted to prove the existence of the Green function, if you've already proved that $\Delta+h$ is an isomorphism between appropriate Sobolev spaces: First you use the procedure in the proof of Theorem 6.5 to construct an approximation to the Green function of the form $\Gamma_0 = r^{2-n}(1+\bar\psi)$; then let $f_0 = (\Delta+h)\Gamma_0$ and note that $f_0$ is in some Sobolev space on which $(\Delta+h)^{-1}$ exists. The Green function is $\Gamma_P = \Gamma_0 - (\Delta + h)^{-1}(f_0)$.
  3. Or you can use the Schwarz kernel theorem to observe that $(\Delta+h)^{-1}$ has an integral kernel $K$, which is a distribution on $M\times M$. Elliptic regularity shows that it is smooth on the complement of the diagonal. The Green function is $\Gamma_P(Q) = K(P,Q)$.

Hope this is helpful.

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=) How cool is this! Thank you very much for these suggestions (and for writing the paper, I guess). –  Sam Feb 20 '13 at 19:46

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