# Understanding the definition of weak solution for the eigenvalue problem of the Colding and Minicozzi's operator

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.

• Have you read the corresponding section in Evans (Colding--Minicozzi's footnote 11)? – Otis Chodosh May 23 at 17:33
• 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? – George May 23 at 23:24
• 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. – Otis Chodosh May 24 at 19:52