Dirichlet Energy
Definition
Let a differentiable function $u : \Omega \to \mathbb{R}$ defined on the open set $\Omega \subset \mathbb{R}^{n}$ be given. The functional defined below is called the Dirichlet energy.
$$ E [u] = \dfrac{1}{2}\int_{\Omega} \|\nabla u(x) \|^{2} \mathrm{d}x $$
Here $\nabla u$ is the gradient of $u$. Gradient
Explanation
Since the gradient is the derivative of a multivariable function, i.e., the rate of change, the Dirichlet energy can be regarded as a quantity that measures how much the function $u$ varies on the given domain $\Omega$. Because the integrand is greater than or equal to $0$, the statement that the Dirichlet energy equals $0$ means $\| \nabla u \|^{2} = 0$, and by the definition of the norm, $\nabla u = 0$ — that is, $u$ is a constant function with no variation.
Properties
- For every $u$, $E [u] \ge 0$.