Integral by parts in Green\s identity and Uniqueness

Question

I am looking at PDE 2nd version written by Evans.On page 42,the theorem uses integral by parts: $ 0=-\int_{U} w\Delta w \,dx = \int_{U} |Dw|^2\,dx $. But when I check Green's identity,I see that one term is missing or equals zero probably:$ \int_{\partial U}w\dfrac{\partial w}{\partial n}\,ds $.Does anyone know why this guy is not here,or maybe show it's $\leq 0$?

Answer 1

Since in the paragraph just before this line, you are told that $w = u - \tilde u$, and from the preceeding page, they both solve $$ v|_{\partial \Omega} = g, $$ then $u-\tilde u$ vanishes on the boundary.