Missing a link in this proof of the Weak Maximum Principle.

Question

In the proof of Theorem 1(b) of this article, the final inequality doesn't seem to follow from Equations (1) and (2). Namely, Equation (2) seems to imply that the inequality $$\sup_{x\in\partial \Omega_+} u \leq \sup_{x\in\partial \Omega} u^+$$ should instead be the equality $$\sup_{x\in\partial \Omega_+} u = \sup_{x\in\partial \Omega} u^+.$$ What am I missing?

Edit: come to think of it, is Theorem 1(a) correct? The proof in that case shows an inequality rather than an inequality. Perhaps the author switched the signs of the two cases by mistake?

Answer 1

You are correct Equation (2) implies an equality not an inequality. But remember in order to get Equation (2) you considered the case $\Omega_+ \neq \emptyset$. If $\Omega_+ = \emptyset$ then $u^+=0$ on $\partial\Omega$ and $\sup_\Omega u \leqslant 0$, so in general you only get an inequality.

For your question about (a), since $ \partial \Omega \subset \overline{\Omega}$, we also have $\sup_{\partial \Omega} u \leqslant \sup_{\Omega} u$. This holds for all $u\in C^0(\overline{\Omega})$ not just subharmonic functions.