Doubt from Maximum principle Elliptic operator

Question

I was reading Maximum principal in PDE form Gilberg Trudinger

Remark

I do understand whole proof but i do not understand remark. I know that maxima of u occure on $\partial \Omega^+$ but How maxima of u occur on $\partial \Omega$ i do not understand

Any help will be appreciated

Answer 1

Hint: if $u\in \partial \bar{\Omega}^+$ then either $u=0$ or $u\in \partial \Omega$.