Spectral decomposition of $-\Delta$ the Laplacian

I am currently trying to self learn about an interesting idea that caught my eye in spectral geometry, which is the whole idea of hearing the shape of the drum by solving the wave equation $\Delta \psi = k^2 \psi_{tt}$ on a compact Riemannian manifold $(M,g)$ with dirichlet boundary constraint $\psi|_{\partial M}=0$. The thing is, I don't think I have that of a solid background in functional analysis and PDEs, and I am looking for some help or a rather self contained source which proves that such decomposition exists $$ -\Delta(\cdot)= \sum_{l=0}^\infty \lambda_l \left \langle \cdot,\phi_l \right \rangle_{L^2(M)}\phi_l$$ What I do know: I wasn't able to find many sources which dive deep into this problem, but the first thing I was able to realize is this decomposition probably isn't on $L^2(M)$ but rather some subspace of the form $V=\left \{ f\ \text{nice enough}|\ f|_{\partial M}=0\right \}$, the reason for this is that we probably need the laplacian to be self adjoint, and looking at one of Green's identities $$\int_M u\Delta v - v\Delta u =\int_{\partial M}u\frac{\partial v}{\partial n}-v \frac{\partial u}{\partial n}$$ We need to zero the RHS with the constraint. What is $V$? I saw alot of references to a "Sobolev space $H_0^1(M)$" but couldn't find a coherent definition of it, in addition to the fact that it is equipped with a different inner product than of $L^2(M)$ which makes things more confusing for me.

Anyhow, this tells us automatically why eigenfunctions must be orthogonal, since if we have $$ -\Delta \phi_1 = \lambda_1 \phi_1,\ -\Delta \phi_2 = \lambda_2 \phi_2$$ for $\lambda_1,\lambda_2 \neq 0$ different then $$\left \langle \lambda_1\phi_1,\phi_2 \right \rangle_{L^2(M)}=-\left \langle \Delta\phi_1,\phi_2 \right \rangle_{L^2(M)}=-\left \langle \phi_1,\Delta\phi_2 \right \rangle_{L^2(M)}=\left \langle \phi_1,\lambda_2\phi_2 \right \rangle_{L^2(M)}$$ and we get that $\left \langle \phi_1,\phi_2 \right \rangle_{L^2(M)}=0$

I also know that $-\Delta$ is positive definite since $$\int_M u (-\Delta) u=\int_M \left \| \nabla u \right \|_g^2\geq 0$$

What I'm looking for: This topic is pretty weird for me right now and I would like to learn about it thoroughly.

1. Are my proofs correct or relevant at all?
2. As I mentioned what is $V$? it's hard for me to see why $C^2(M) \cap C(\bar{M})$ needs to be restricted further.
3. How do we know that $-\Delta$ is compact if at all? This means that if $f_k \rightarrow f$ in $V$, then $-\Delta f_k$ has a convergent subsequence of functions. I know according to functional analysis this should imply the spectral decomposition but I don't know whether this is true. Edit: appearantly not, $-\Delta^{-1}$ is. Then I am looking at the moment for a proof of that
4. Assuming 3 is true, we also get for free that the eigenvalues are discrete $0\leq \lambda_0 \leq \lambda_1 \leq \cdots$. However, I saw claims that $\lambda_k \rightarrow \infty$ (when according to functional analysis it should approach to $0$). This doesn't make sense to me, how sums like the spectral decomposition $\Delta f=\sum_{l=0}^\infty \lambda_l\left \langle f,\phi_l \right \rangle_{L^2(M)}\phi_l $ converge?
5. How does one show $\left \{ \phi_l\right \}_{l=0}^\infty$ is complete? meaning we can express any $f\in V$ as a fourier series $f=\sum_{l=0}^\infty \left \langle f,\phi_l \right \rangle_{L^2(M)}\phi_l $

Sorry for the mess of a question, In the probable case I said something wrong I would love to hear. If anyone knows of a free-access source which adresses this problem please refer me to it!

EDIT: I have read several notes and articles which address this issue. Right now, it all boils down for me to find proofs of the Rellich-Kondrachov theorem and Poincaré inequality for Riemannian manifolds, and I can take it from there using the theory of weak solutions.

Solution 1:

A confession: I can only really answer this when the manifold in question is an open set $U$ of $\mathbb{R}^n$, where $\partial U$ is smooth. (I would imagine this case is used, or at least very instructive for, the case on a general Riemannian manifold.) But I'll try to answer in this setting as best I can. I apologize in advance for the length of this answer.

2) The first thing that must be done is to understand what $V$ is, because functional analytic machinery needs to be applied on a particular (Hilbert) space. Indeed, in this case $V = H^1_0(U)$. I don't know if there is any way to get around having to deal with this object, but you can think of $H^1_0(U)$ as the space of all function $f$ which are zero on the boundary (this is what the "$0$" in the subscript refers to), are square-integrable, and whose derivatives exist and are also square-integrable. This is a Hilbert space. The inner product is not just the $L^2$ inner product anymore, because the inner product on $L^2$ does not care about the derivatives, but our space does. So we define the inner product by $$ \langle f, g \rangle = \int_U f(x)g(x)\, dx + \int_U \nabla f(x) \cdot \nabla g(x)\, dx. $$ Note that $\langle f, g \rangle = \langle f, g\rangle_{L^2} + \langle \nabla f, \nabla g\rangle_{L^2}$. The last thing that must be done is to note that this space is not complete if we restrict to the usual defintion of derivatives, and therefore the formal definition is that $f, g$ have square-integrable "weak" derivatives. This is morally perhaps unimportant, but important in the rigorous proofs. This is essentially why you cannot take $V = C^2(U) \cap C^2(\bar{U})$, since this space is not complete under the norm induced by the inner product above, and so if we take a limit of functions in this space, the result won't necessarily be $C^2$.

1) Your calculations are essentially correct, but can be (and in the general theory are) extended to functions with these "weak" derivatives. It turns out not to be important for your calculations involving $\phi_i$ which are eigenvalues of $-\Delta$, since elliptic regularity actually implies all eigenfunctions of $-\Delta$ are smooth.

3) This is where the full machinery comes in. As rubikscube09 mentioned in the comments, I don't believe there is a way to do this without the Rellich-Kondravich theorem (all of this is in Chapter 5, 6 of Evans) and abstract functional analysis. Essentially, the R-K theorem says that the embedding $H^1_0(U) \to L^2(U)$ is compact, in that bounded sequences in $H^1_0(U)$ have an $L^2$-convergence subsequence. Therefore we may think of $(-\Delta)^{-1}$ as mapping from $L^2$ to $H^1_0$ defined by $(-\Delta)^{-1}f = u$ is the unique function such that $-\Delta u = f$. Then $u$ lives in $H^1_0(U)$ (and again, there is something being swept under the rug, since $u \in H^1_0(U)$ only implies $u$ has first order derivatives, not second-order), but the embedding $H^1_0(U) \to L^2(U)$ identifies $u$ as an $L^2$ function in a compact way and thus we may think of $(-\Delta^{-1}) : L^2 \to L^2$. By R-K this is a compact operator.

4) You addressed this in your comment.

5) This is a computation that is a little involved, but not too long. I don't know of free-source materials, but Evans ch. 6 deals with this (in more generality) and has detailed proofs there. There may be copies online?

Solution 2:

After a lot of digging online I have found the following resources very useful

• The Calderón problem on Riemannian manifolds by Mikko Salo
• PDEs Basic Theory by Michael Taylor

There they treat with no lack of rigour this exact problem, in the spirit of @Chris' answer. Leaving it here since it might help anyone who sees this in the future.