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  "body": "\n## Problem\n\nLet $n \\ge 2$ and let $\\Omega \\subseteq \\R^n$ be a bounded connected [[open set|02BH3MCBQB]] whose [[boundary|05D5EGMFYV]] $\\partial\\Omega$ is a $C^1$ hypersurface, with outward unit normal $\\nu$.\nLet $g \\colon \\partial\\Omega \\to \\R$ be a [[continuous function|0X3WGX95FE]].\nFind a function $u$ that is twice continuously differentiable on $\\Omega$, continuously differentiable on $\\overline{\\Omega}$, [[harmonic|0DEB0JMN0W]] on $\\Omega$, and satisfies the [[Neumann boundary condition|GS3KT2H645]]\n$$\\frac{\\partial u}{\\partial \\nu} = g \\quad \\text{on } \\partial\\Omega.$$\n\n## Remarks\n\nA necessary condition is $\\int_{\\partial\\Omega} g \\, dS = 0$: by the [[divergence theorem|0JX86PSSA9]], $\\int_{\\partial\\Omega} \\partial u / \\partial \\nu \\, dS = \\int_\\Omega \\Delta u \\, dx = 0$.\nA solution is unique up to an additive constant: the difference $w$ of two solutions satisfies $\\int_\\Omega \\abs{\\nabla w}^2 \\, dx = \\int_{\\partial\\Omega} w \\, \\partial w / \\partial \\nu \\, dS = 0$, so $w$ is constant on the connected set $\\Omega$.\nFor bounded domains with sufficiently smooth boundary, for example $C^2$, a solution exists whenever the condition $\\int_{\\partial\\Omega} g \\, dS = 0$ holds; one proof uses layer potentials and Fredholm theory.\nExample: on the unit disc with $g(\\theta) = \\cos\\theta$, the condition holds and $u(x, y) = x$ is a solution, since $\\partial u / \\partial \\nu = x = \\cos\\theta$ on the unit circle.\nThe problem is the companion of the [[Dirichlet problem|8D56GQ4AG8]], which prescribes $u$ instead of its normal derivative.\n\n## Explanations\n\n### L1 human\n\nImagine a metal body in thermal equilibrium, where you control how much heat flows in or out through each piece of its surface.\nThe Neumann problem asks for the temperature inside.\nIn equilibrium the total inflow must be zero, and the temperature is then determined up to adding a constant.\n",
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