I am trying to understand the corollary $5.15$ on page $23$ of the paper GENERIC MEAN CURVATURE FLOW I; GENERIC SINGULARITIES by Colding and Minicozzi. Specifically, I would like to understand why they stated the eigenvalue problem for the operator $L$, which is defined at the bottom of the page $21$, with weak solutions in the weighted $L^2$ space instead of the weighted $W^{1,2}$ space, which sounds more natural to me. I believe that the key for the answer is that $L$ is self-adjoint with respect the inner product of the weighted $L^2$ space by the theorem $5.2$ on page $22$, but I can not see how this can justify that the solutions live in the weighted $L^2$ space instead of the weighted $W^{1,2}$ space and I do not think that these weighted spaces are equal once that the respective spaces without the weight are not equal.
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$\begingroup$ Have you read the corresponding section in Evans (Colding--Minicozzi's footnote 11)? $\endgroup$– Otis ChodoshMay 23, 2020 at 17:33
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$\begingroup$ Yes, I have read, but I could not adapt the proof of the theorems in Evans' book for the Colding and Minicozzi's operator, then I followed the section $3$ of this lecture notes: math.mcgill.ca/gantumur/math580f18/laplacenotes.pdf and I forgot to come back to Evans' book when my doubt arised. Anyway, coming back to Evans' book, it seems clear now, the solutions live in the weighted $W^{1,2}$ space, but the spectral theorem states that the solutions are an orthonormal basis to the weighted $L^2$ space. Is this, isn't it? $\endgroup$– GeorgeMay 23, 2020 at 23:24
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$\begingroup$ In fact the solutions are completely smooth (by elliptic regularity)! But they are a orthonormal basis of $L^2$. Of course this seems a bit weird, but you should remember, for example that polynomials are dense in $L^2[0,1]$, for example. So it is not so strange that you can find an orthonormal basis of $L^2$ that consists of smooth functions. $\endgroup$– Otis ChodoshMay 24, 2020 at 19:52
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