there. This is part of the textbook of Gibarg's PDE:
My question is that how to verify the part in red?
How to know $\overline w$ is continous in $\overline \Omega$?
Thanks so much! Your help will be appreciated! :)
2026-03-27 14:21:14.1774621274
Local barrier implies barrier?
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since $m=\inf_{N-B}w\gt 0$, on $\partial B$, $m\le w$, and so $\overline{w}|_{\partial B}=m$, as $w$ is superharmonic it is continuous. The $\inf$ of two continous functions is continuous, thus $\overline{w}$ is continuous.