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  "body": "\n## Definition\n\nLet $n \\ge 2$ and let $\\Omega \\subseteq \\R^n$ be an [[open set|02BH3MCBQB]].\nA *positive harmonic function* on $\\Omega$ is a [[harmonic function|0DEB0JMN0W]] $u \\colon \\Omega \\to \\R$ such that $u(x) \u003e 0$ for every $x \\in \\Omega$.\n\n## Remarks\n\nExample: on the upper half-plane $\\set{(x, y) : y \u003e 0}$, the function $u(x, y) = y$ is positive harmonic, since $\\Delta u = 0$ and $y \u003e 0$ there.\nNon-example: $u(x, y) = x$ on the same half-plane is harmonic but takes negative values, for instance at $(-1, 1)$.\nSome authors call a harmonic function with $u \\ge 0$ nonnegative harmonic; on a connected open set, the minimum principle shows that such a function is either identically $0$ or positive everywhere.\nOn a ball, positive harmonic functions satisfy [[Harnack's inequality|YNH832D50W]].\n",
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