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Oct 19, 2015 at 8:57 comment added Delio Mugnolo Are you missing the explanation of why the solution is $W^{2,p}$ in the very first place? Well, it also follows directly from Thm. 2.4.2.6 from the same book.
Oct 19, 2015 at 6:50 vote accept Thanasis Stylianou
Oct 19, 2015 at 6:50 comment added Thanasis Stylianou It is not the complete reference but together with Theorem 7.1 of Giaquinta's book provides us with what we need. Thanks, I was well aware of Grisvard's books but I somehow "missed" the section on smoother domains :-)
Oct 19, 2015 at 6:35 history answered Delio Mugnolo CC BY-SA 3.0