Definition of regularity in PDE theory I'm studying a course in PDE theory and we talked a lot about "regularity". However, we never gave a precise definition about what "regularity of weak solutions" is. 
Also a search on the web didn't clarify for me what regularity exactly is. Somehow I think that we want to find out how smoothly the weak solutions of a PDE behave, i.e. if a solution is $C^\infty$ for instance. 
My question is: Can someone pin down what people think about, when they talk about "regularity of weak solutions"?
Thanks!
 A: Regularity of weak solutions generally means in which space they are, most of the times we use Sobolev space to mention the regularity, for example we say function is in $H^1$ space , to mean that its first derivative is in $L^2$, and so on..
A: Regularity is one of the vague yet very useful terms to talk about a vast variety of results in a uniform way. Other examples of such words include "dynamics" in dynamical systems (I have never seen a real definition of this term but everyone uses it, and it vaguely means the way a system changes over time) or "canonical" (roughly meaning that with just the information given, a canonical choice is the most obvious choice) in more algebraic contexts.
In general, more regularity means more desirable properties.
Typically this means one or several of the following:


*

*Higher integrability, i.e. a function lies in a more restrictive $L^p$ space, i.e. higher powers of the function are integrable

*Higher differentiability, i.e. higher (weak) derivatives of the function exist.


Furthermore, the issue of regularity is often isolated in proofs, so e.g. if you want to prove existence and uniqueness of a classical solution (i.e. "strongly" differentiable function that actually satisfies a PDE in exactly the way that it is posed) to a PDE, you first show existence and uniqueness of a weak solution which is often much easier (and allows to use tools from, for example, Hilbert space theory which are not available in most cases when looking for strong solutions). And only then you proceed to show that this solution is actually much more regular, for example, it actually possesses not only weak but even strong derivatives. A typical way to achieve this is using the Sobolev embedding theorems which tell you that if a function has sufficiently many weak derivatives, or rather, sufficiently regular weak derivatives (i.e. lies in a Sobolev space $W^{k,p}$ with large enough $k$ and large enough $p$), then it can be identified with a function that is actually strongly differentiable. This statement is called the "second part of the Sobolev embedding theorem" in the article linked above and is nicely summarised in the statement
$$
W^{k,p}(\mathbb{R}^{n}) \subset C^{r, \alpha}(\mathbb{R}^{n})
$$
under suitable conditions on the "regularity coefficients" $k,p$ (and the dimension $n$). 
Note also that these results are not limited to the full space $\mathbb{R}^{n}$, but also to domains $\Omega \subset \mathbb{R}^{n}$ that satisfy certain conditions (which, funnily, are also often referred to as "regularity" conditions, for example that the domains boundary $\partial \Omega$ can be represented by a "smooth enough" function, say a function in $C^{1}$, and then we write $\partial \Omega \in C^{1}$).
In this way, you can sometimes solve PDEs in the classical sense by making a detour into the realm of weak solutions.
Hope that helps!
EDIT: A nice example of this "weak + regular $\Rightarrow$ strong" at work can be found in the answer to this post:
When can one expect a classical solution of a PDE?
