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Let $M$ be a compact Riemannian manifold (without boundary), I would like to know under which regularity conditions can we solve the heat equation $$\begin{align} \partial_tu-\Delta u &= f \\ u(\cdot,0) &= u_0(\cdot) \end{align}$$ in $M\times[0,T]$?

I have just read this question which seems to imply that we can solve the equation when $f\in C^{0,0,\alpha}((0,T)\times M,\mathbb{R})$ and $u_0\in C^{2,\alpha}(M,\mathbb{R})$. I would like to know what is the general method of solving when we have this kind of regularity.

The way I learned about solving heat equation in Euclidean domain was by using heat kernel for unbounded domain, and Galerkin approximation on a bounded domain $\Omega$ with boundary. For the Galerkin method, we use the fact that the eigenvectors of the isomorphism $$ -\Delta:H^2(\Omega)\cap H^1_0(\Omega)\to L^2(\Omega) $$ form an orthogonal basis in $L^2(\Omega)$. This seems to rely on the fact that $u\equiv 0$ on $\partial\Omega\times[0,T]$, I don't know if similar result holds on compact manifold without boundary.

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  • $\begingroup$ I think you can solve the heat equation in $L^2$ by taking the eigenfunctions and evolving only the eigenvalues. $\endgroup$ – Ben McKay Mar 29 '18 at 10:21
  • $\begingroup$ @BenMcKay Does the set of eigenvectors of $-\Delta$ also forms an orthogonal basis when our domain is a compact manifold without boundary? I am sorry if this is a silly question but I have very little experience in geometric analysis. $\endgroup$ – BigbearZzz Mar 29 '18 at 10:29
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For a brief overview, See Gallot, Hulin, Lafontaine, Riemannian Geometry, Springer, 1st edition, p. 198, theorem 4.43, for the spectral theorem for compact Riemannian manifolds: the eigenfunctions are smooth and dense in $L^2$, and the eigenspaces are finite dimenional, so we can pick an orthonormal basis of smooth eigenfunctions. Unfortunately, they give no proof, but they have some references, and I think the proofs are in Colin de Verdiere, Quasi-modes sur les varietes Riemanniennes, Inv.Math., 43 (1977), 15-52.

Edit: complete proofs are given in Olivier Lablée, Spectral Theory in Riemannian Geometry, European Mathematical Society, 2010, ; see Theorem 4.3.1. p. 77.

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  • $\begingroup$ Unfortunately I cannot read French, but thank you very much for the reference. $\endgroup$ – BigbearZzz Mar 29 '18 at 11:30
  • $\begingroup$ You find this also in Shubins book, Theorem 8.3, and I would assume it is in every book containing the phrases pseudodifferential operators or spectral theory. $\endgroup$ – mcd Mar 29 '18 at 14:39

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