Dirichlet Boundary Conditions 📂Partial Differential Equations

Dirichlet Boundary Conditions

Definition¹

Let us assume that a partial differential equation is given on an open set $\Omega$. The following boundary conditions are referred to as Dirichlet boundary conditions. The problem of finding solutions to partial differential equations with Dirichlet boundary conditions is called the Dirichlet problem.

$$ u = 0 \quad \text{on } \partial \Omega $$

Explanation

Nonhomogeneous Conditions

The following boundary conditions are referred to as nonhomogeneous Dirichlet conditions, although, in many cases, there is no meticulous distinction made between homogeneous and nonhomogeneous.

$$ u = g \quad \text{on } \partial \Omega $$

Example

For instance, solving the Dirichlet problem for Poisson’s equation involves finding $u$ that satisfies the following.

$$ \left\{ \begin{align*} -\Delta u = f & \quad \text{in } \Omega \\ u = 0 & \quad \text{on }\partial \Omega \end{align*} \right. $$