I am studying BV space (the space of functions of bounded variation) by using Evans & Gariepy's book. However, there are no exercises for BV space, and I have no idea where I can find some.
Could you guys recommend me a book which has plenty exercise in BV space, as well as SBV space? (I know Luigi Ambrosio, Nicola Fusco, Diego Palla has some, but I wish to see more).
For a gentle introduction to functions of bounded variations, I recommend A First Course in Sobolev Spaces By Leoni, where Chapter 2 concerns one-variable BV space and Chapter 13 deals with several variables. There are many exercises throughout the text.
Special functions of bounded variation (SBV) is a recent topic, not even 30 years old yet. The book that you mention (Functions of bounded variation and free discontinuity problems by Ambrosio, Fusco, Pallara) is the only comprehensive treatment of SBV that I know of, written by people who developed much of the subject. There are later books that use SBV as a tool, but they are more specialized and I would not expect you to find exercises there.