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.

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.
