Dirichlet boundary condition

Definition

Let $a < b$, and consider a second-order ordinary differential equation for $y$ on $[a, b]$. A Dirichlet 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, with boundary $\partial \Omega$, and let $g \colon \partial \Omega \to \R$ be a given function. For an equation for $u$ on $\Omega$, the Dirichlet boundary condition is $u(x) = g(x)$ for all $x \in \partial \Omega$. The condition is homogeneous if $\alpha = \beta = 0$, respectively $g = 0$.