View source for Dirichlet boundary conditions

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''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|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|First boundary value problem]]).

See also [[Second boundary value problem|Second boundary value problem]]; [[Neumann boundary conditions|Neumann boundary conditions]]; [[Third boundary value problem|Third boundary value problem]].