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I'm currently reading some papers by Uhlenbeck on harmonic maps. She mentions the following facts:

Let $M^m$ and $N^n$ be compact Riemannian manifolds, $N$ embedded isometrically into Euclidean space.

1) For $p > 1$ the space $W^{1,p}(M,N)$ is a Banach manifold.

2) For $p>m$ the space $W^{1,p}(M,N)$ is a $C^\infty$ Banach manifold.

3) For $p>m$ the $p$-energy functional $\int_M|du(x)|^pd\mu$ satisfies the Palais-Smale condition.

The reference that she gives, Palais' Foundations of Global Nonlinear Analysis is too daunting a read at this time for me. There seem to be no other references where these facts are proved. If anyone could give a proof of any of these three ( especially 2) or 3) ), it would be much appreciated. A reference other than Palais' would also be much appreciated.

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    $\begingroup$ Yet I'd suggest to make an effort and try reading Palais' FNGA, which is a model of mathematical writing for style and clearness. $\endgroup$ Oct 16, 2014 at 6:36
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    $\begingroup$ R. Palais turns out to be an active member of MO. $\endgroup$ Oct 16, 2014 at 8:40
  • $\begingroup$ I've read Palais' papers on Morse Theory on Hilbert manifolds, and Lyusternik–Schnirelmann theory of Banach manifolds. They are some of the best written papers I've seen. I completely agree with you Pietro, and I have made the effort. However, FNGA stresses a very general, highly functorial view of the matters. It would simply take me too far a field to read up to page 125 of that paper where the PS condition is proven. The literature seems to imply that these results are well known, that is the only reason I assume there would be other references. $\endgroup$
    – student
    Oct 16, 2014 at 14:24
  • $\begingroup$ But I do plan on reading FNGA one day. It's just not possible for me at this time, with my workload. $\endgroup$
    – student
    Oct 16, 2014 at 14:45

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