Dirichlet boundary conditions

Dirichlet conditions, Dirichlet data, boundary conditions of the first 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 form

$$u(x)=\phi(x),\quad x\in S,$$

are called Dirichlet boundary conditions.

A boundary value problem with Dirichlet conditions is also called a boundary value problem of the first kind (see First boundary value problem).

See also Second boundary value problem; Neumann boundary conditions; Third boundary value problem.