Dirichlet Energy

Give a region \(\Omega\) and a function \(u\), Dirichlet energy is the integral of square of gradient in the region:

\begin{align*} E_D(u) = \frac 1 2 \int_{\Omega} \left|\nabla u\right|^2 dA \end{align*}

In a way Dirichlet energy measurse the smoothness of a function. If we consider constant function as the smoothest possible function. Dirichlet energy measure the total deviation/failure of the function to be a constant function.

If we want to find the function that minimizes Dirichlet Energy. Solution to Laplace equation gives the solution.

\begin{align*} &u^* = \min_{u:\Omega \rightarrow \mathbb{R}} E_D(u) \\ &\nabla E_D(u^*) = 0 \end{align*}

Since \(E_D\) is convex, if we find the minima, we know it is the global minima. This ties in with the uniqueness of solution to Laplace Equation.

Since solution to laplace equation are harmonic function, they are also the minimzer of dirichlet energy.