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Sorry for the bump: I plan to start an area 51 site on something closely related, namely community reviews: meta.mathoverflow.net/q/4320/40644 Any comments or support is welcome!
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 :-)
Gilbarg and Trudinger provide the result as a product of the Schauder theory which assumes Hölder regularity of the boundary. It seems that this is also the case with Han and Lin
@Math604 Yes, this is what I had in mind, it should work similarly to the Neumann problem for the Laplacian (I did not do all the details here so let me know if you encounter any difficulties...).
