Neumann problem

Remarks

A necessary condition is $\int_{\partial\Omega} g \, dS = 0$: by the divergence theorem, $\int_{\partial\Omega} \partial u / \partial \nu \, dS = \int_\Omega \Delta u \, dx = 0$. A 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 $\Omega$. For 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. Example: 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. The problem is the companion of the Dirichlet problem, which prescribes $u$ instead of its normal derivative.