Verfiying a proof in PDE Theory I am a student in Computer Science and I have gain an interest in PDE Theory.  I got a book on this subject authored by L.C. Evans. In the book problems are solved in a analytic way. I was wondering how is verified that a proof is correct. Could someone provide me some insight? Are their books on how to verify the correctness of a proof in this subject? The reason for asking this question is because it does not always make sense that something is true. This implicates that I pretend  understanding the theory, rather than understanding the theory.
 A: In general, proofs are not completely rigorous, in the sense that some details are always left to the reader (e.g. not every minute step of a long calculation is included, we don't need to recall the properties of real numbers every time we want to multiply two of them, etc.). A good author knows their audience, and writes with this in mind; in Evans' PDE book, he assumes (for instance) that you know some basic facts from calculus, and so he doesn't burden the exposition by including a bunch of unneeded details. In a more advanced text, the author might assume that you already know the contents of Evans' PDE, and so they will skip those sorts of details in order to explore even more advanced topics without this additional level of (again, burdensome) detail.
For the purposes of checking correctness of a proof, this is basically done by the author's peers; certainly any new results (to be published in a paper) go through a referee process, and then even after the paper is published, if any errors snuck through, they are likely to be noticed eventually. In the case of writing a book, new results are typically not included, and so you can feel reasonably "safe" regarding the correctness of the arguments (all the stated results should have been established and verified before the book was written), and again, any errors sneaking into the text should be caught eventually and fixed in errata/subsequent editions.
In your own study, it is often the case that something "obvious" left unsaid is not at all obvious to you as a reader. In this case, I think it is usually safe to assume it is correct, and then work toward understanding why. This is a completely natural and common situation for you to find yourself in. Think about it on your own, ask your peers or mentors, post a request for clarification to stackexchange, that's all part of the process of reading mathematics. 
(as a post-script, there are of the ideas of computer-assisted proofs/proof-verification, but this is not the standard in mathematical writing, at least not at present).
A: Modern formal proofs are written using proof assistants, not by hand.
The general agreement in mathematics is that writing formal proofs is time-consuming and not particularly insightful for the reader, so people use more accessible language (as in Evan's book), with the underlying assumption that the arguments can be made formal.
I am not aware of particular efforts to proof-check classical PDE arguments. Some searching yields a particular piece of work focused on a formal proof of a numerical scheme's convergence.
