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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\R^n$ be a bounded open set with $C^1$ boundary $\\partial\\Omega$, outward unit normal $\\nu$ and surface measure $dS$.\nFor a function $w$ of class $C^1$ near $\\partial\\Omega$, write $\\partial w / \\partial \\nu = \\nabla w \\cdot \\nu$ for its normal [[derivative|06KWGHDN2J]].\nLet $u$ and $v$ be twice continuously differentiable on an open set that contains the closure $\\overline{\\Omega}$.\n\n1. (First identity.) $\\displaystyle \\int_\\Omega \\left( u \\, \\Delta v + \\nabla u \\cdot \\nabla v \\right) dx = \\int_{\\partial\\Omega} u \\, \\frac{\\partial v}{\\partial \\nu} \\, dS.$\n2. (Second identity.) $\\displaystyle \\int_\\Omega \\left( u \\, \\Delta v - v \\, \\Delta u \\right) dx = \\int_{\\partial\\Omega} \\left( u \\, \\frac{\\partial v}{\\partial \\nu} - v \\, \\frac{\\partial u}{\\partial \\nu} \\right) dS.$\n3. (Third identity.) For every $x \\in \\Omega$,\n$$u(x) = \\int_{\\partial\\Omega} \\left( \\Phi(y - x) \\, \\frac{\\partial u}{\\partial \\nu}(y) - u(y) \\, \\frac{\\partial \\Phi}{\\partial \\nu_y}(y - x) \\right) dS(y) - \\int_\\Omega \\Phi(y - x) \\, \\Delta u(y) \\, dy,$$\nwhere $\\Phi$ is the [[fundamental solution of the Laplacian|BSJ8X2XRN9]], normalised by $-\\Delta\\Phi = \\delta$.\n\nHere $\\Delta$ is the [[Laplacian|0E0B6RA8K0]] and $\\nabla$ is the [[gradient|0ABK9V9Y64]].\n\n## Remarks\n\nThe first identity is the [[divergence theorem|0JX86PSSA9]] applied to the vector field $u \\nabla v$, whose divergence is $u \\Delta v + \\nabla u \\cdot \\nabla v$.\nThe second identity follows by writing the first one also with $u$ and $v$ exchanged and subtracting.\nThe third identity applies the second one with $v(y) = \\Phi(y - x)$ on $\\Omega$ minus a small ball $B(x, \\varepsilon)$, and lets $\\varepsilon \\to 0$; the small sphere contributes $u(x)$ in the limit.\nFor the first identity it is enough that $u$ is of class $C^1$.\nExample: with $u = 1$, the first identity gives $\\int_\\Omega \\Delta v \\, dx = \\int_{\\partial\\Omega} \\partial v / \\partial \\nu \\, dS$; so a harmonic function has zero total flux through $\\partial\\Omega$.\nExample: with $u = v$ harmonic, it gives $\\int_\\Omega \\abs{\\nabla u}^2 dx = \\int_{\\partial\\Omega} u \\, \\partial u / \\partial \\nu \\, dS$, which proves uniqueness for the Dirichlet problem.\n\n## Explanations\n\n### L1 human\n\nGreen's identities are the integration by parts formulas for the Laplacian.\nThey trade an integral over a region for an integral over its boundary.\nIn one dimension the first identity is just $\\int_a^b (u v'' + u' v') \\, dx = [u v']_a^b$.\n\n### L2 human\n\nThese identities are the basic tool for boundary value problems of Laplace and Poisson type.\nThe first one gives energy estimates and uniqueness of solutions, and it underlies the weak formulation of elliptic equations.\nThe second one shows that the Laplacian is symmetric for functions that vanish on the boundary, which leads to the symmetry of Green's functions.\nThe third one is the representation formula: a function is determined inside $\\Omega$ by its Laplacian, its boundary values and its normal derivatives.\nIt leads to the Poisson integral formula once a Green's function is known.\n",
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