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In our lecture on PDE, we were told that Sobolev Spaces are very important (but not how and why). Now, I am wondering why they are so important and what I need them for in the context of PDE?

I've seen these posts: Significance of Sobolev spaces for numerical analysis & PDEs? and What do we need Sobolev-spaces $W^{k, p}$ with $p \neq 2$ for? and there it is mentioned that Sobolev Spaces are needed for weak solutions. Unfortunately, I still have trouble to understand what is stated in those posts... are my weak solution always in Sobolev Spaces (and only in Sobolev Spaces)? Are there any other benefits from these spaces if I look at elliptic differential operators in particular?

It would be great if someone could explain this to me in fairly simple terms :)

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(This is just my opinion on Sobolev spaces.)

At the end of the day, the idea behind PDE theory is to understand how solutions of PDEs behave. One of the first questions you need to answer to even get started is whether the solutions have discontinuities or other singularities. One way to measure how big of a singularity a function has is to look at its integrability using an $L^p$ norm. A function that is in $L^p$ with large $p$ can't have a too-significant spike, and if it is in $L^\infty$ then the function has to be even bounded.

Another measure of the "badness" of a function is to look at how differentiable it is. This can't be detected by looking at integrability because even $L^\infty$ functions can have discontinuities. Thus it's natural to incorporate derivatives into function norms. A simple way to use both of these notions of regularity is to define the $W^{k,p}$ spaces.

One way to think about the statement that $f\in W^{k,p}$ is that it is a summary of what we know about $f$. It's a very incomplete summary, but it turns out that it's enough in many cases to be able to make sense of nonlinear or multilinear expressions involving $f$ and other functions.

In summary, I'd say that Sobolev spaces are useful because they are the simplest spaces one can define that are still strong enough to allow you to do the manipulations needed to work with the complicated expressions arising in PDE. Moreover there is a mature theory associated with them, so one can often combine off-the-shelf bounds to prove powerful theorems.

What I hope is clear from this is that Sobolev spaces are not always the most natural tool for a particular problem. Sometimes more refined information is needed, and sometimes one only has control on certain combinations of derivatives (rather than a blanket bound on all possible partial derivatives).

To get a better sense of the relationship between the spaces, I'd recommend looking at a map like https://terrytao.wordpress.com/2010/03/11/a-type-diagram-for-function-spaces/ .

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