Neumann boundary condition

Definition

Let $a < b$, and consider a second-order ordinary differential equation for $y$ on $[a, b]$. A Neumann boundary condition is a pair of conditions $$y'(a) = \alpha, \qquad y'(b) = \beta,$$ with given real numbers $\alpha, \beta$. More generally, let $\Omega \subseteq \R^d$ be a bounded domain, that is, a connected open set, whose boundary $\partial \Omega$ is of class $C^1$. Let $\nu(x)$ be the outward unit normal vector at $x \in \partial \Omega$, and let $g \colon \partial \Omega \to \R$ be a given function. For an equation for $u$ on $\Omega$, the Neumann boundary condition is $$\frac{\partial u}{\partial \nu}(x) = \nabla u(x) \cdot \nu(x) = g(x) \quad \text{for all } x \in \partial \Omega,$$ where $\frac{\partial u}{\partial \nu}$ is the directional derivative of $u$ along $\nu$.