Neumann boundary conditions

Neumann conditions, Neumann data, boundary conditions of the second kind

Consider a second-order partial differential equation $Lu=f$ on a domain $D$ in $\mathbf R^n$ with boundary $S$ (cf. also Differential equation, partial, of the second order). Boundary conditions of the type

$$\frac{\partial u(x)}{\partial n}=\phi(x),\quad x\in S,$$

where $n$ is the outward pointing normal at $x$, are called Neumann boundary conditions.

A boundary value problem with Neumann conditions is also called a boundary value problem of the second kind (see Second boundary value problem).

See also First boundary value problem; Dirichlet boundary conditions; Third boundary value problem.