In mathematics, the Dirichlet boundary condition is imposed on an ordinary or partial differential equation, such that the values that the solution takes along the boundary of the domain are fixed.
The question of finding solutions to such equations is known as the Dirichlet problem.
[1] In finite-element analysis, the essential or Dirichlet boundary condition is defined by weighted-integral form of a differential equation.
[2] The dependent unknown u in the same form as the weight function w appearing in the boundary expression is termed a primary variable, and its specification constitutes the essential or Dirichlet boundary condition.
the Dirichlet boundary conditions on the interval [a,b] take the form
denotes the Laplace operator, the Dirichlet boundary conditions on a domain Ω ⊂ Rn take the form