What should I know about Sobolev space?

Question

I have done some Sobolev spaces with some embedding theorems, trace theorems etc. Sorry that my question is really vague.

If my professor asks me what is great about Sobolev space, what should I answer (details, examples, counterexamples are very much welcome) to make sure that he feels ok this guy knows the concept pretty well.

Any readings, insights, examples, counterexamples are most welcome.

Thanks for your help.

Answer 1

For my experience in learning Sobolev space, you probably need to know the theorems, and how to approach the theorems, not necessary to be very detailed, the idea is enough.

As an example, for embedding theorem, remembering the content is boring, but you have to know the meaning of this theorem, it tells us we can sacrifices the regularity[But we cannot do the reverse usually] to obtain more integrability, which is extremely useful in PDE theories.

Terry's blog has a note for Sobolev space, it has some very good exercises for Sobolev spaces, you can take them as examples.

Answer 2

n many problems of mathematical physics and variational ca lculus it is not sufficient to deal with the classical solutions of differentia l equations. It is necessary to introduce the notion of weak derivatives and to work in the so called Sobolev spaces

http://www.iadm.uni-stuttgart.de/LstAnaMPhy/Weidl/fa-ws04/Suslina_Sobolevraeume.pdf