Dirichlet energy

Summary

The Dirichlet energy of a function $u$ on a domain is $\frac12 \int \abs{\nabla u}^2 \, dx$, half the integral of the squared length of its gradient. It measures how much the function varies. Its critical points with fixed boundary values are the harmonic functions, which is the content of Dirichlet's principle. The energy is the quadratic form behind the Laplacian, the Sobolev space $H^1$ and the theory of Dirichlet forms.