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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $\\Omega \\subseteq \\R^n$ be a bounded [[Lipschitz domain|1GNKQ7NBAG]].\nThen there are constants $r_0 \u003e 0$ and $C \\ge 1$, depending only on $\\Omega$, with the following property.\nLet $\\zeta \\in \\partial\\Omega$ and $0 \u003c r \u003c r_0$, and let $u, v$ be positive [[harmonic functions|0DEB0JMN0W]] on $\\Omega \\cap B(\\zeta, 2r)$ that extend continuously by $0$ to $\\partial\\Omega \\cap B(\\zeta, 2r)$.\nThen\n$$\\frac{u(x)}{v(x)} \\le C\\, \\frac{u(y)}{v(y)} \\qquad \\text{for all } x, y \\in \\Omega \\cap B(\\zeta, r).$$\n\n## Remarks\n\nThe constant $C$ does not depend on $\\zeta$, $r$, $u$ or $v$; it depends only on $n$ and on the Lipschitz character of $\\Omega$.\nInside $\\Omega$, the ordinary [[Harnack inequality|YNH832D50W]] compares the values of one positive harmonic function at nearby points.\nThe boundary version compares two functions up to the boundary, where both tend to $0$: their ratio stays bounded above and below.\nIdea of the proof: a Carleson-type estimate bounds $u$ near the boundary by its value at an interior point at distance about $r$ from $\\zeta$.\nComparison with harmonic measure, using that the Lipschitz graph satisfies an exterior cone condition at every scale, then gives matching lower bounds.\nExample, a local check in the upper half-plane near $\\zeta = 0$: take $u(x, y) = y$ and $v(x, y) = y / ((x - 3)^2 + y^2)$, a constant multiple of the Poisson kernel with pole $3$.\nOn $B(0, 1)$ the ratio $v / u = 1 / ((x - 3)^2 + y^2)$ lies between $1/16$ and $1/4$, so the conclusion holds with $C = 4$.\nThe vanishing hypothesis is needed: with $u = y$ and $v = 1$, the ratio $u / v = y$ tends to $0$ at the boundary, and no constant works.\n\n## Explanations\n\n### L1 human\n\nTwo positive harmonic functions that both vanish on a piece of the boundary vanish there at the same rate.\nSo near that piece, one is roughly a constant multiple of the other.\nThis holds even when the boundary has corners, as long as it is a Lipschitz graph.\n\n### L2 human\n\nThe principle is the key estimate in the boundary theory of harmonic functions on rough domains.\nIt implies that ratios of positive harmonic functions vanishing on a boundary portion are Hölder continuous up to the boundary.\nFrom this one finds the Martin boundary of a Lipschitz domain: it equals the Euclidean boundary, and every boundary point is minimal.\nIt also underlies the comparison of harmonic measure with the Green function and the doubling property of harmonic measure on Lipschitz domains.\n",
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