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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\R^n$ be a bounded domain, that is, a bounded connected open set.\nLet\n$$L u = \\sum_{i,j=1}^n a^{ij}(x) \\, \\partial_i \\partial_j u + \\sum_{i=1}^n b^i(x) \\, \\partial_i u + c(x) \\, u$$\nbe a uniformly [[elliptic|B4TTNHSTRX]] operator in $\\Omega$ with bounded coefficients, $a^{ij} = a^{ji}$, and $c \\le 0$ in $\\Omega$.\nLet $u, v \\in C^2(\\Omega) \\cap C^0(\\overline{\\Omega})$ satisfy $L u \\ge L v$ in $\\Omega$ and $u \\le v$ on $\\partial\\Omega$.\nThen $u \\le v$ in $\\Omega$.\n\n## Remarks\n\nProof idea: the function $w = u - v$ satisfies $L w \\ge 0$ in $\\Omega$ and $w \\le 0$ on $\\partial\\Omega$.\nThe weak [[maximum principle|D2JF19HF91]] for operators with $c \\le 0$ gives $\\sup_\\Omega w \\le \\sup_{\\partial\\Omega} w^+ = 0$.\n\nExample: if $\\Delta u \\ge 0$ in the unit ball $B$ and $u \\le 1$ on $\\partial B$, comparing with $v = 1$ gives $u \\le 1$ in $B$.\nIf $-\\Delta u = 1$ in $B \\subseteq \\R^n$ with $u = 0$ on $\\partial B$, comparison with $0$ and with $(1 - \\abs{x}^2)/(2n)$ shows $0 \\le u \\le (1 - \\abs{x}^2)/(2n)$; in fact $u$ equals this function.\n\nThe sign condition on $c$ is needed: for $L u = u'' + u$ on $(0, \\pi)$, the functions $u = \\sin x$ and $v = 0$ satisfy $L u = L v = 0$ and $u = v$ at the endpoints, but $u \u003e v$ inside.\n\nUniqueness for the Dirichlet problem $L u = f$ in $\\Omega$, $u = g$ on $\\partial\\Omega$, follows by applying the theorem twice.\n\n## Explanations\n\n### L1 human\n\nIf two solutions of the same kind of equation are ordered on the boundary, they stay ordered inside.\nSo to bound an unknown solution, it is enough to find an explicit function that dominates it on the boundary and satisfies the right inequality inside.\nSuch functions are called barriers, or super- and subsolutions.\n\n### L2 human\n\nThe comparison principle is the main use of the maximum principle.\nIt gives uniqueness for boundary value problems, a priori bounds by explicit barriers, and monotone iteration schemes between a subsolution and a supersolution.\nVersions exist for parabolic equations, through the [[parabolic maximum principle|EMDVD4E83K]], and for fully nonlinear equations, where comparison for [[viscosity solutions|H1ERJ3T75T]] is the key uniqueness tool.\nIt is the PDE analogue of the [[comparison theorem|7XTBR347P5]] for ordinary differential equations.\n",
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  "name": "Comparison principle",
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