Barrier

Remarks

Example: suppose there is a closed ball $\overline{B}(y, R)$ with $\overline{B}(y, R) \cap \overline{\Omega} = \set{\xi}$, and let $n \ge 3$. Then $w(x) = R^{2-n} - \abs{x - y}^{2-n}$ is a barrier at $\xi$. It is continuous on $\overline{\Omega}$ and harmonic on $\Omega$, since $y \notin \overline{\Omega}$; it vanishes at $\xi$ because $\abs{\xi - y} = R$; and $\abs{x - y} > R$ for the other points $x \in \overline{\Omega}$, which gives $w(x) > 0$. For $n = 2$ the function $w(x) = \log(\abs{x - y}/R)$ plays the same role. Non-example: $w(x) = \abs{x - \xi}^2$ satisfies clauses 1 and 3, but $\Delta w = 2n > 0$, so it is strictly subharmonic and fails clause 2. A local barrier is a function with these properties on $\overline{\Omega} \cap N$ for some neighbourhood $N$ of $\xi$. A local barrier yields a barrier, so the two notions give the same regular points.